POLYNOMIAL ROOT DISTRIBUTION AND ITS IMPACT ON SOLUTIONS TO
THUE EQUATIONS
by
GREG KNAPP
A DISSERTATION
Presented to the Department of Mathematics
and the Division of Graduate Studies of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2023
DISSERTATION APPROVAL PAGE
Student: Greg Knapp
Title: Polynomial Root Distribution and Its Impact on Solutions to Thue Equations
This dissertation has been accepted and approved in partial fulfillment of the requirements
for the Doctor of Philosophy degree in the Department of Mathematics by:
Shabnam Akhtari Chair
Ellen Eischen Core Member
Weiyong He Core Member
Ben Young Core Member
Brittany Erickson Institutional Representative
and
Krista Chronister Vice Provost for Graduate Studies
Original approval signatures are on file with the University of Oregon Division of
Graduate Studies.
Degree awarded June 2023.
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© 2023 Greg Knapp
3
DISSERTATION ABSTRACT
Greg Knapp
Doctor of Philosophy
Department of Mathematics
June 2023
Title: Polynomial Root Distribution and Its Impact on Solutions to Thue Equations
In this study, we focus on two topics in classical number theory. First, we examine
Thue equations—equations of the form 𝐹 (𝑥, 𝑦) = ℎ where 𝐹 (𝑥, 𝑦) is an irreducible, inte-
gral binary form and ℎ is an integer—and we give improvements to both asymptotic and
explicit bounds on the number of integer pair solutions to Thue equations. These improved
bounds largely stem from improvements to a counting technique associated with “The Gap
Principle,” which describes the gap between denominators of good rational approximations
to an algebraic number. Next, we will take inspiration from the impact of polynomial root
distribution on solutions to Thue equations and we examine polynomial root distribution as
its own topic. Here, we will look at the relation between the separation of a polynomial—
the minimal distance between distinct roots—and the Mahler measure of a polynomial—a
height function which connects the roots of a polynomial with its coefficients. We make
a conjecture about how separation can be bounded above by the Mahler measure and we
give data supporting that conjecture along with proofs of the conjecture in some low-degree
cases.
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ACKNOWLEDGMENTS
I would like to thank my advisor, Shabnam Akhtari, for her help in selecting interesting
and accessible problems and for all of her support throughout the challenges of graduate
school. She showed me, above all, how to be an incredible mentor.
I would like to thank the others who contributed to the mathematics contained in this
dissertation. Ben Young gave helpful feedback on the code that went into optimizing the
parameters 𝑇 and 𝑍 . JoeWebster gave me the regular reminder to look at as many examples
as possible and he inspired much of the progress that I made on relating Mahler measure
and separation. Chris Sinclair gave me specific suggestions to enable my work on Mahler
measure and separation. This research was supported by NSF grant number 2001281.
I would like to thank my more general mathematical role models. Cathy Hsu was
incredibly supportive throughout my time in graduate school, from helping me find a social
group in my first year, to helping me get myself organized in my fourth year, to helping
me with my job application materials in my sixth year (among many other things). Ellen
Eischen set a great example of what a successful researcher and effective teacher looks like.
Joe Webster always modeled the value of being a clear expositor and teacher. Ken Barrett
gave me the drive to open up every black box and ask why the metaphorical machinery is
constructed the way it is.
I would like to thank my therapist for regularly reminding me about the importance
of balance. He helped me make the last six years bearable (and more than that, fun) and
showed me that the best dissertation is a done dissertation.
I would like to thank my family and friends for continuously being there for me. Jake
Anderson, Kyle Bahnsen, Marissa Masden, Johanna Meven, Kelly Pohland, Austin Ray,
Melissa Ruszczyk, Joseph Tate, Steven Tokar, Michael Volkovitsch, and Eli Wolff have
consistently been able to pull me away from my work to play games and I’m so happy that
they try so hard to do that. Samantha Platt asked hard questions and helped me reflect on so
many issues I didn’t know I was taking for granted. Alex Roesch was the one person outside
of math who was genuinely excited about my research in pure mathematics. Gabrielle
Cohen helped me stay in shape and reminded me that there’s a lot that happens outside of
the math department. Jamie Piatka was my confidante long before graduate school and I
am grateful for her steadfast presence in my life. Kyla Pohl has been an incredible partner
and she does a great job cheering me up after tough days. My parents have continued to be
role models and have done so much to help me grow into a more mature person.
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TABLE OF CONTENTS
Chapter Page
TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1. Using Approximation to Categorize Real Numbers . . . . . . . . . . . 11
1.2. (In)effectiveness of Approximation Results . . . . . . . . . . . . . . . 15
1.3. Thue Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.1 Connecting Thue Equations to Diophantine Approximation . . . . . 16
1.3.2 New Results on Thue Equations . . . . . . . . . . . . . . . . . . . 19
1.4. Root Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4.1 Number of Real Roots . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4.2 Roots in a Compact Region . . . . . . . . . . . . . . . . . . . . . . 23
1.4.3 Separation of Polynomial Roots . . . . . . . . . . . . . . . . . . . 24
1.4.4 New Results on Polynomial Root Separation . . . . . . . . . . . . . 27
2. THUE EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2. Conjectures and Heuristics . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3. Notation and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4. Previous Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4.1 Large Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.2 Medium Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.3 Small Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5. Technical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.6. Asymptotic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.6.1 Improved Bounds on the Number of Medium Solutions to Thue’s
Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.6.2 Proofs of Lemmas 2.23 and 2.24 . . . . . . . . . . . . . . . . . . . 45
2.6.3 Improving Akhtari and Bengoechea’s Medium Solution Bound . . . 50
2.6.4 Proof of Theorem 1.16 . . . . . . . . . . . . . . . . . . . . . . . . 52
2.7. Explicit Results for Trinomials . . . . . . . . . . . . . . . . . . . . . . 53
2.7.1 Small Special Solutions . . . . . . . . . . . . . . . . . . . . . . . . 55
2.7.2 Large Special Solutions . . . . . . . . . . . . . . . . . . . . . . . . 59
2.7.3 Choosing Parameters for Large Degrees . . . . . . . . . . . . . . . 64
6
2.7.4 Choosing Parameters for Small Degrees . . . . . . . . . . . . . . . 68
2.7.5 Attaining Bounds With Examples . . . . . . . . . . . . . . . . . . . 70
3. ROOT SEPARATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.2. Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.2.1 Lower Bounds on Separation . . . . . . . . . . . . . . . . . . . . . 73
3.2.2 Upper Bounds on Separation . . . . . . . . . . . . . . . . . . . . . 75
3.3. Data and Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.4. Proof of Theorem 1.34 . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
APPENDICES
A. PARAMETER CHOICES FOR SECTION 2.7.4 . . . . . . . . . . . . . . . . . . 96
B. PYTHON AND SAGE CODE . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
B.1. Python Code for Section 2.7.4 . . . . . . . . . . . . . . . . . . . . . . . 112
B.2. Sage Method For Section 2.7.5 . . . . . . . . . . . . . . . . . . . . . . 115
B.3. Sage Methods for Section 3.3 . . . . . . . . . . . . . . . . . . . . . . . 116
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7
LIST OF FIGURES
Figure Page
3.1. Separation against Mahler measure for monic quadratic polynomials with two
real roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.2. Separation against Mahler measure for monic quadratic polynomials with no
real roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.3. Separation against Mahler measure for monic cubic polynomials with three real
roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.4. Separation against Mahler measure for monic cubic polynomials with one real
root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.5. Logarithmic separation against logarithmicMahlermeasure formonic quadratic
polynomials with no real roots . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.6. Logarithmic separation against logarithmicMahlermeasure formonic quadratic
polynomials with two real roots . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.7. Logarithmic separation against logarithmic Mahler measure for monic cubic
polynomials with three real roots . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.8. Logarithmic separation against logarithmic Mahler measure for monic cubic
polynomials with one real root . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.9. Logarithmic separation against logarithmic Mahler measure for monic quartic
polynomials with four real roots . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.10. Logarithmic separation against logarithmic Mahler measure for monic quartic
polynomials with two real roots . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.11. Logarithmic separation against logarithmic Mahler measure for monic quartic
polynomials with no real roots . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.12. Mahler measure against separation for monic quadratic polynomials with two
real roots and the sharp upper bound of 𝑦 = 𝑥 + 1. . . . . . . . . . . . . . . . . 87
3.13. Mahler measure against separation for monic quadratic polynomials with no
real roots and the sharp upper bound of 𝑦 = 2𝑥1/2. . . . . . . . . . . . . . . . . 87
3.14. Mahler measure against separa√︃tion for monic cubic polynomials with 3 real√ √ √
roots and the upper bound 𝑦 = 10 𝜋/(3 7) ∗ 𝑥. . . . . . . . . . . . . . . . 90
3.15. Mahler measure against separation√︁for monic quartic polynomials with 4 real√
roots and the upper bound 𝑦 = log( 2 𝜋/3) + 𝑥/2. . . . . . . . . . . . . . . . 90
3.16. Mahler measure against separation√for monic cubic polynomials with only one
real root and the upper bound 𝑦 = 3𝑥. . . . . . . . . . . . . . . . . . . . . . . 93
8
3.17. Logarithmic Mahler measure against logarithmic separation √for monic quartic
polynomials with no real roots and the upper bound 𝑦 = log( 2) + 𝑥/4 . . . . . 95
9
LIST OF TABLES
Table Page
2.1. Summary of parameter choices which minimize 𝑇 + 𝑍 . . . . . . . . . . . . . . 70
2.2. The maximal number of solutions to equation (2.4) for a trinomial 𝐹 (𝑥, 𝑦) of
degree 𝑛 and height 𝐻 ⩽ 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.3. The maximal number of solutions to equation (2.4) for a trinomial 𝐹 (𝑥, 𝑦) of
degree 𝑛 and height 𝐻 ⩾ 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.1. Parameter choices which minimize 𝑇 + 𝑍 . . . . . . . . . . . . . . . . . . . . . 111
10
CHAPTER 1
INTRODUCTION
The main thesis of our study is that properties of polynomial roots can be understood
through knowledge about the coefficients of the polynomial. It is well known that the the
roots of a polynomial of degree at least five cannot necessarily be expressed in an
elementary way in terms of the coefficients of that polynomial, but the inability to express
the roots of the polynomial exactly does not prohibit us from understanding certain
features of the roots.
One of the interesting properties we can ask about is which rational numbers are close
to the roots of a polynomial. The question of finding good rational approximations to real
numbers has been thoroughly explored through the field of Diophantine approximation,
which we describe in the introduction. Chapter 2 will discuss an application of
Diophantine approximation to the study of Thue equations and we will prove some new
results bounding the number of solutions to Thue equations.
Another question we can ask about the roots of a polynomial is how they are
distributed in the complex plane. In Chapter 3, we discuss previous work finding lower
bounds on the distance between roots of polynomials in terms of the coefficients and we
turn that question around to ask about upper bounds on the distances between roots. We
show how those upper bounds provide us with some ability to quantify the statement “the
roots of polynomials are not randomly distributed,” we conjecture what the sharpest upper
bounds might look like, and we prove sharp upper bounds in some low-degree settings.
1.1 Using Approximation to Categorize Real Numbers
We begin our exploration of Diophantine approximation by showing how
approximation results can be used to classify real numbers. One of the foundational
theorems in this area is Dirichlet’s Approximation Theorem.
Theorem 1.1 (Dirichlet). Let 𝛼 ∈ R and 𝑄 ∈ Z>0. Then there exist integers 𝑝, 𝑞 ∈ Z with
1 ⩽ 𝑞 ⩽ 𝑄 so that
|𝑞𝛼 − | 1𝑝 < .
𝑄
Dirichlet’s proof of this theorem is clever [Dir42].
Proof. For any real number 𝑥, let {𝑥} denote the fractional part of 𝑥, namely
{𝑥} := 𝑥 − ⌊𝑥⌋ . Subdivide the half-open unit interval [0, 1) into 𝑄 intervals of the form
11
[ )
𝐽 = 𝑛−1𝑛 , 𝑛 . Now consider the sequence of 𝑄 + 1 numbers 0, {𝛼}, {2𝛼}, . . . , {𝑄𝛼}.𝑄 𝑄
These 𝑄 + 1 numbers all lie in [0, 1) and by the pigeonhole principle (also known as
Dirichlet’s box principle), there exist 𝑛, 𝑠1, 𝑠2 ∈ N with 0 ⩽ 𝑠1 < 𝑠2 ⩽ 𝑄 so that
{𝑠1𝛼}, {𝑠2𝛼} ∈ 𝐽𝑛. Hence,
|{𝑠1𝛼} − {𝑠2𝛼}|
1
< .
𝑄
But writing {𝑠1𝛼} = 𝑠1𝛼 − 𝑟1 and {𝑠2𝛼} = 𝑠2𝛼 − 𝑟2 for integers 𝑟1 and 𝑟2 yields
| (𝑠2 − 𝑠1)𝛼 − (𝑟2 − 𝑟1) |
1
<
𝑄
and so we take 𝑞 = 𝑠2 − 𝑠1 and 𝑝 = 𝑟2 − 𝑟1. We note that the fact that 1 ⩽ 𝑞 ⩽ 𝑄 follows
from the fact that 0 ⩽ 𝑠1 < 𝑠2 ⩽ 𝑄. □
On the face of it, it is not immediately obvious that Dirichlet’s Theorem has anything
to do with approximation. However, it has the following immediate corollary. Before
stating the corollary, we give a quick definition.
Definition 1.2. A pair (𝑝, 𝑞) ∈ Z2 is primitive if gcd(𝑝, 𝑞) = 1.
Corollary 1.3. Let 𝛼 ∈ R. Then 𝛼 is irrational if and only if there are infinitely many
primitive pairs (𝑝, 𝑞) ∈ Z2 with 𝑞 > 0so that − 𝑝𝛼  1< 2 . (1.1)𝑞 𝑞
Proof. Suppose first that 𝛼 is rational, so write 𝛼 = 𝑟 for integers 𝑟 and 𝑠. Note that for
𝑠
any integers 𝑝 and 𝑞 with 𝑝 ≠ 𝑟 = 𝛼, w e have𝑞 𝑠   𝑝 𝑟𝑞 − 𝑠𝑝 −  1𝛼 = ⩾𝑞 𝑞𝑠 |𝑞𝑠 |
since |𝑟𝑞 − 𝑠𝑝 | is a nonzero, positive integer. Hence, if the primitive pair (𝑝, 𝑞) satisfies
inequality (1.1), then it must satisfy
1 1
| <𝑞𝑠 | 𝑞2
,
implying that |𝑞 | < |𝑠 |. Then there are only finitely many possible values for 𝑞. It is now
easy to check that for each 𝑞 there are only finitely many possible values of 𝑝 that satisfy
(1.1) and hence, there are only finitely many primitive pairs (𝑝, 𝑞) that satisfy (1.1).
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Next, suppose that 𝛼 is irrational. Suppose by contradiction that there are only finitely
many primitive pairs (𝑝1, 𝑞1), . . . , (𝑝𝑛, 𝑞𝑛) satisfying (1.1). Observe first that because 𝛼 is
irrational,
𝑚 := min |𝑞𝑖𝛼 − 𝑝𝑖 | > 0.
1⩽𝑖⩽𝑛
Then set
+ ⌈ 1𝑄 := 1 ⌉
𝑚
so that
1
< |𝑞𝑖𝛼 − 𝑝𝑖 |
𝑄
for all 1 ⩽ 𝑖 ⩽ 𝑛. Apply Dirichlet’s Theorem to this value of 𝑄 to find 𝑝, 𝑞 ∈ Z so that
1 ⩽ 𝑞 ⩽ 𝑄 and for all 1 ⩽ 𝑖 ⩽ 𝑛,
| 1𝑞𝛼 − 𝑝 | < < |𝑞𝑖𝛼 − 𝑝𝑖 |.
𝑄
Writing 𝑑 = gcd(𝑝, 𝑞) and setting 𝑝′ = 𝑝/𝑑 and 𝑞′ = 𝑞/𝑑, we see that the pair (𝑝′, 𝑞′) is
primitive and we have ′ = 𝑞𝑞 ⩽ 𝑄. Further,
𝑑
|𝑞′𝛼 − 𝑝′| ⩽ 𝑑 |𝑞′𝛼 − 1𝑝′| = |𝑞𝛼 − 𝑝 | <
𝑄
implying that  ′ 𝑝𝛼 − 𝑞′  1 1< 𝑄𝑞′ ⩽ .(𝑞′)2
Hence, the pair (𝑝′, 𝑞′) satisfies (1.1). Moreover, (𝑝′, 𝑞′) is distinct from all of the pairs
(𝑝 , 𝑞 ) because |𝑞′𝛼 − 𝑝′𝑖 𝑖 | < |𝑞𝑖𝛼 − 𝑝𝑖 |. This contradicts the hypothesis that we had listed
all pairs satisfying (1.1); hence, there must be infinitely many primitive pairs satisfying
(1.1). □
Note that the “if” direction of the corollary can actually be improved somewhat: 𝛼 is
irrational if for any 𝜀 > 0, there are infinitely many primitive pairs (𝑝, 𝑞) ∈ Z2 with 𝑞 ≠ 0
so that |𝛼 − 𝑝 | < 1
𝑞 𝑞1+𝜀
.
Corollary 1.3 gives our first indication that the classification of 𝛼 as rational or
irrational can be encoded in the language of rational approximation. The first question one
might ask about this corollary is if it can be improved. The answer is yes, but only by a
constant factor. This combines Theorem 5B with Lemma 2E in Chapter I.2 of [Sch80].
13
Proposition 1.4. Let 𝛼 ∈ R. Then 𝛼 is irrational if and only if there are infinitely many
primitive pairs (𝑝, 𝑞) ∈ Z2 with 𝑞 > 0 so that 𝛼 − 𝑝  1< √ .𝑞 5𝑞2
√
Moreover, the golden ratio 𝜑 = 1+ 52 is irrational, yet for any constant 𝐶 < √
1 , there
5
are only finitely many primitive primitive pair 
s (𝑝, 𝑞) ∈ Z2 with 𝑞 > 0 so that
𝑝
𝜑 − 
𝑞  𝐶< 𝑞2 .
There is another sense in which Corollary 1.3 can be somewhat improved. Observe
that since irrational numbers comprise almost all of the real line according to Lebesgue
measure, Corollary 1.3 implies that for almost all real numbers 𝛼, there are infinitely many
primitive pairs (𝑝, 𝑞) ∈ Z2 with 𝑞 > 0 so that (1.1) holds. However, Khinchin in [Khi64]
shows that
Theorem 1.5 (Khinchin). For almost all real numbers 𝛼, there are infinitely many
primitive pairs (𝑝, 𝑞) ∈ Z2 with 𝑞 > 0 so that − 𝑝𝛼  1<𝑞 𝑞2 .log 𝑞
From these results, it is natural to ask what sort of information we can learn about
rational approximations if we start with finer hypotheses about the algebraicity of 𝛼.
Liouville’s Theorem on approximation is one of the first along these lines. This is
Theorem 1A of Chapter V.1 in [Sch80].
Theorem 1.6 (Liouville). Suppose that 𝛼 ∈ R is algebraic∗ of degree 𝑛. Then there exists
a constant 𝐶 (𝛼) > 0 so that for any p rimitive pair (𝑝, 𝑞) ∈ Z
2 with 𝑞 > 0,
 − 𝑝  𝐶 (𝛼)𝛼 > .𝑞 𝑞𝑛
Liouville’s Theorem indicates that a real algebraic 𝛼 of degree 𝑛 cannot be
approximated by infinitely many rationals according to the law | − 𝑝𝛼 | < 1
𝑞 𝑞𝑛+𝜀 when 𝜀 is
any positive real number. However, Liouville’s Theorem can be radically improved, as
Roth showed in [Rot55].
∗Of course, Liouville’s Theorem also holds for algebraic 𝛼 ∉ R, but we will generally
not focus on this case in this chapter.
14
Theorem 1.7 (Roth). Suppose that 𝛼 ∈ R is algebraic and 𝜀 > 0. Then there are only
finitely many primitive pairs (𝑝, 𝑞) ∈  2Z with 𝑞 > 0 so that 𝑝 1𝛼 −  <𝑞 𝑞2+ . (1.2)𝜀
In summary, we note that if 𝛼 is rational, then for any 𝐶 > 1, there are infinitely many
rationals 𝑝 which satisfy | 𝑝𝛼 − | < 𝐶 . However, there are only finitely many rationals
𝑞 𝑞 𝑞
which satisfy | − 𝑝𝛼 | < 𝐶1+𝜀 . If 𝛼 is an irrational algebraic number, then for any 𝐶 > 1,𝑞 𝑞
there are infinitely many rationals 𝑝 which satisfy |𝛼 − 𝑝 | < 𝐶2 . However, there are only𝑞 𝑞 𝑞
finitely many rationals 𝑝 which satisfy | 𝑝𝛼 − | < 𝐶
𝑞 𝑞 𝑞2+𝜀
.
Based on this pattern, one might expect that if 𝛼 is transcendental, then there would be
infinitely many solutions to some inequality like |𝛼 − 𝑝 | < 𝐶3 . However, a law like this is𝑞 𝑞
dramatically false. Theorem 32 in Khinchin’s [Khi64] implies that
Theorem 1.8 (Khinchin). For almost all real numbers 𝛼, there are only finitely many
primitive pairs (𝑝, 𝑞) ∈ Z2 with 𝑞 > 0 so that − 𝑝 𝛼 𝑞  1< .𝑞2(log 𝑞)1+𝜀
So while the theory of rational approximation is currently able to completely
distinguish between rational and irrational numbers, we do not currently have a way of
completely distinguishing between algebraic and transcendental numbers.
1.2 (In)effectiveness of Approximation Results
Note that the proof of Corollary 1.3 is effective for rational 𝛼 in the sense that it gives a
method for finding the finitely many primitive pairs (𝑝, 𝑞) which can satisfy (1.1): the size
of 𝑞 can be bounded in terms of the denominator of 𝛼 and for each 𝑞, one can easily bound
the size of 𝑝 in terms of 𝑞 and 𝛼. Hence, there are not only finitely many primitive pairs
(𝑝, 𝑞) satisfying (1.1), they all live in a finite search space which can be easily expressed
in terms of 𝛼. Liouville’s Theorem is also effective in the sense that 𝐶 (𝛼) can be
expressed in terms of 𝛼 (see Theorem 6.1 in [Eve21]). However, Roth’s Theorem is highly
ineffective: the proof gives no method of finding the finitely many solutions to (1.2).
One of the main goals of modern Diophantine approximation is then to find effective
improvements of Liouville’s Theorem. Fel’dman manages to improve Liouville’s general
theorem in an effective way.
15
Theorem 1.9 (Fel’dman). Let 𝛼 be algebraic of degree 𝑛 ⩾ 3. Then there exist effectively
computable constants 𝐶 (𝛼) > 0 and 𝑎(𝛼) > 0 so that for any primitive pair (𝑝, 𝑞) ∈ Z2with 𝑞 > 0,  − 𝑝  𝐶 (𝛼)𝛼 >𝑞 𝑞𝑛−𝑎(𝛼) .
In [Fel71], Fel’dman notes that he plans to estimate the sizes of 𝐶 (𝛼) and 𝑎(𝛼) in a
future paper, but no such paper later appears, nor do other authors appear to take up this
task. As it stands then, we shall have to be satisfied with knowing that 𝐶 (𝛼) and 𝑎(𝛼) can
be computed with enough patience.
That said, even if it is quite difficult to improve upon Fel’dman’s result about all
rational approximations to 𝛼 in an effective way, there are effective methods which can
improve upon Fel’dman’s result in certain settings. In particular, solutions to Thue
equations provide a natural setting where the quality of the corresponding rational
approximations can be effectively measured.
1.3 Thue Equations
1.3.1 Connecting Thue Equations to Diophantine Approximation
Of course, before we can see how solutions to Thue equations give rise to good
rational approximations of algebraic numbers which can be effectively described, we must
first discuss what a Thue equation is. Our first definition will help us concisely state the
hypotheses we regularly assume:
Definition 1.10. An integral binary form is a homogeneous polynomial in two variables
with integer coefficients.
Next, we define what a Thue equation is.
Definition 1.11. Let 𝐹 (𝑥, 𝑦) ∈ Z[𝑥, 𝑦] be an irreducible integral binary form of degree
𝑛 ⩾ 3 and let ℎ ∈ Z. Then the equation
𝐹 (𝑥, 𝑦) = ℎ (1.3)
is known as a Thue equation.
A major number-theoretic goal is to find all of the solutions to (1.3) in integers. In this
document, whenever we refer to a solution to a Thue equation, we specifically mean a pair
16
of integers (𝑝, 𝑞) for which 𝐹 (𝑝, 𝑞) = ℎ. The solutions to Thue equations tend to produce
good rational approximations to the roots of the one-variable polynomials 𝐹 (𝑥, 1) and
𝐹 (1, 𝑦). For instance, by dehomogenizin(g, co)nsider that a solution (𝑝, 𝑞) ∈ Z
2 to
𝐹 (𝑥, 𝑦) = ℎ with 𝑞 ≠ 0 yields
𝑝 ℎ
𝐹 , 1 = .
𝑞 𝑞𝑛
Writing 𝑓 (𝑥) = 𝐹 (𝑥, 1) and factoring 𝑓 (𝑥) o∏ver C[𝑥] as𝑛
𝑓 (𝑥) = 𝑏 (𝑥 − 𝛼𝑖),
𝑖=1
we note that this implies that
| | ∏𝑛ℎ
= |𝑏 | | |  𝑝 − 𝛼𝑖𝑞 𝑛 𝑞 
𝑖=1
and hence, for sufficiently large 𝑞, 𝑝 gives a good rational approximation of some root 𝛼
𝑞 𝑖
of 𝑓 (𝑥). Likewise, by reversing the roles of 𝑥 and 𝑦, one can see that for sufficiently large
𝑝, 𝑞 gives a good rational approximation of some root 𝛼∗ of 𝐹 (1, 𝑦).
𝑝 𝑖
The rational approximations to a root 𝛼 of 𝐹 (𝑥, 1) which arise from solutions to Thue
equations tend to satisfy much stronger approximation laws than a generic rational
approximation to 𝛼. Moreover, those laws tend to include effective constants, like the
height of the polynomial 𝑓 (𝑥). The following definition is the only definition of height
that we will use throughout this paper. However, it is worth observing that this height is
the naïve height of a polynomial and this is related to, but not identical to, other heights
such as the Weil height.
Definition 1.12. Given a polynomial 𝑔(𝑥1,∑︁. . . , 𝑥𝑛) ∈ C[𝑥1, . . . , 𝑥𝑛] of the form
𝑖
𝑔(𝑥 1 𝑖1, . . . , 𝑥𝑛) = 𝑎 𝑛𝑖1,...,𝑖 𝑥 · · · 𝑥 ,𝑛 1 𝑛
𝑖1,...,𝑖𝑛
the height of 𝑔(𝑥1, . . . , 𝑥𝑛) is
𝐻 (𝑔) = max |𝑎𝑖1,...,𝑖 |.𝑛
𝑖1,...,𝑖𝑛
The height of a polynomial with integer coefficients gives a sense of its complexity:
the larger the height, the more bits of information required to represent the polynomial.
The height of a single-variable polynomial 𝑓 (𝑥) is a useful, effectively computable
constant often found in results about approximations of the roots of 𝑓 (𝑥). Consider this
result from Bombieri and Schmidt in [BS87] on the rational approximations one can
obtain from solutions to Thue equations:
17
Proposition 1.13 (Bombieri and Schmidt). Suppose that (𝑝, 𝑞) ∈ Z2 is a solution to the
Thue equation 𝐹 (𝑥, 𝑦) = ℎ, where 𝑛 ⩾ 3 denotes the degree of 𝐹 and 𝑝, 𝑞 ≠ 0. Then there
exists a root 𝛼 of 𝐹 (𝑥, 1) with(  )− 𝑝  ((2𝑛 + 2)𝐻 (𝐹))𝑛 ℎmin 1, 𝛼 ⩽ | | . (1.4)𝑞 𝑞 𝑛
By symmetry, there exists a ro(ot 𝛽 of 𝐹 ( 
1), 𝑦) with
− 𝑞  ((2𝑛 + 2)𝐻 (𝐹))𝑛 ℎmin 1, 𝛽 ⩽ | | . (1.5)𝑝 𝑝 𝑛
Note that the exponents on |𝑞 | and |𝑝 | in this theorem are far better than we would
expect from something like Dirichlet’s Theorem. In fact, Roth’s Theorem guarantees that
there can be only finitely many rational numbers satisfying inequalities (1.4) and (1.5)†
and hence, finitely many integer-pair solutions to the Thue equation 𝐹 (𝑥, 𝑦) = ℎ.‡
It was Mahler’s realization in [Mah33] that inequalities like (1.4) can actually provide
bounds on the number of solutions to the Thue equation 𝐹 (𝑥, 𝑦) = ℎ. Mahler did not give
any kind of method for finding the solutions, but instead found bounds for the number of
good rational approximations to the roots of 𝐹 (𝑥, 1) and 𝐹 (1, 𝑦) and translated those into
bounds on the number of solutions to the Thue inequality 𝐹 (𝑥, 𝑦) = 1. From there Mahler
was able to estimate the number of solutions to 𝐹 (𝑥, 𝑦) = ℎ. This exploration instigated
one of the major projects in the study of Thue equations: finding good bounds on the
number of solutions to 𝐹 (𝑥, 𝑦) = ℎ, which is the topic we analyze in chapter 2.
Given the large size of the constant factor ((2𝑛 + 2)𝐻 (𝐹))𝑛ℎ, Proposition 1.13 only
gives tight bounds on the quality of the approximation (and hence, the number of solutions
to 𝐹 (𝑥, 𝑦) = ℎ) when |𝑞 | is large. For smaller values of 𝑞, Mueller and Schmidt in [MS88]
are able to improve the size of the constant at the cost of reducing the exponent on |𝑞 |:
Proposition 1.14 (Mueller and Schmidt). Suppose that (𝑝, 𝑞) ∈ Z2 is a solution to the
Thue equation 𝐹 (𝑥, 𝑦) = ℎ where 𝑛 ⩾ 3 denotes the degree of 𝐹 (𝑥, 𝑦), 𝑠 + 1 denotes the
†Roth’s Theorem only guarantees that there are finitely many rationals with |𝛼− 𝑝/𝑞 | <
((2𝑛 + 2)𝐻 (𝐹))𝑛ℎ/|𝑞 |𝑛 and in fact, it is not hard to see that there are infinitely many
rationals satisfying (1.4). However, those infinitely many rationals have bounded |𝑞 | and
hence unbounded |𝑝 |. Only finitely many of those rationals can then satisfy (1.5).
‡Both Roth’s Theorem and Bombieri and Schmidt’s lemma came rather later than Thue’s
realization that the equations bearing his name have only finitely many solutions, but they
provide a quick proof here so we will not worry overmuch about this.
18
number of nonzero coefficients of 𝐹 (𝑥, 𝑦), and 𝑞 ≠ 0 with |𝑞 | larger than an explicit
constant depending only on 𝐹 and ℎ.§ Then there is a set 𝑆 of roots of 𝐹 (𝑥, 1) and a set 𝑆∗
of roots of 𝐹 (1, 𝑦), both with cardinalities less than or equal to 6𝑠 + 4 so that either 𝑝𝛼 −  𝐾 (𝐹, ℎ)<𝑞 |𝑞 |𝑛/𝑠
for some 𝛼 ∈ 𝑆 or 𝛼∗ − 𝑞  𝐾 (𝐹, ℎ)<𝑝 |𝑝 |𝑛/𝑠
for some 𝛼∗ ∈ 𝑆∗.
In Mueller and Schmidt’s proposition, the value of 𝐾 (𝐹, ℎ) is explicit and it is
meaningfully smaller than the constant ((2𝑛 + 2)𝐻 (𝐹))𝑛ℎ that appears in Bombieri and
Schmidt’s proposition. Moreover, 𝐾 (𝐹, ℎ) is a multiple of a negative power of 𝐻 (𝐹),
indicating that for polynomials with large height, solutions to the corresponding Thue
equation must produce particularly good rational approximations of the roots of 𝐹 (𝑥, 1)
and 𝐹 (1, 𝑦).
The role of the parameter 𝑠 is not apparent, but it will be explained in chapter 2.
However, it is worth noting that 𝑠 can be small while 𝑛 is large, so the Mueller and
Schmidt’s proposition still provides a good exponent on |𝑞 | (or |𝑝 |, depending on which is
the denominator) and it has a smaller constant than Bombieri and Schmidt’s proposition.
1.3.2 New Results on Thue Equations
In chapter 2, we will explore how this parameter 𝑠 impacts the number of solutions to
Thue equations. In particular, we will improve bounds on the number of solutions to
general Thue equations given by Mueller and Schmidt in [MS88], and Saradha and
Sharma in [SS17]. The following theorem is our main asymptotic¶ result. Before we state
the result, however, we introduce some notation that will be useful throughout the
remainder of this paper.
§This constant is smaller than the constant needed to make Bombieri and Schmidt’s
result useful.
¶Our use of the word “asymptotic” does not always indicate that some parameter is
going to infinity. Rather, we use the word “asymptotic” to refer to a bound where we only
give the main term and we disregard error terms and constants.
19
Notation 1.15. For any set 𝑋 and functions 𝑓 : 𝑋 → R and 𝑔 : 𝑋 → R, the notation
𝑓 (𝑥) ≪ 𝑔(𝑥) means that there exists a constant 𝐶 > 0 so that 𝑓 (𝑥) ⩽ 𝐶𝑔(𝑥) for all 𝑥 ∈ 𝑋 .
Sometimes, we will also write 𝑓 (𝑥) = 𝑂 (𝑔(𝑥)) to mean 𝑓 (𝑥) ≪ 𝑔(𝑥) or we may simply
write 𝑂 (𝑔(𝑥)) to refer to some function 𝑓 (𝑥) which satisfies 𝑓 (𝑥) ≪ 𝑔(𝑥). If 𝑓 (𝑥) and
𝑔(𝑥) both depend on some parameter 𝑛, then the notation 𝑓 (𝑥) ≪𝑛 𝑔(𝑥) means that there
exists a constant 𝐶 > 0 which may depend on 𝑛 so that 𝑓 (𝑥) ⩽ 𝐶𝑔(𝑥) for all 𝑥 ∈ 𝑋 . In
these cases, when using big-oh notation, we will write 𝑂𝑛 (𝑔(𝑥)) to indicate that the
implicit constant depends on 𝑛.
The following theorem makes use of the parameter Φ, which will be defined at the
beginning of Section 2.3. For now, it is only important to know that it satisfies
log3(𝑠) ≪ 𝑒Φ ≪ 𝑠.
Theorem 1.16. Let 𝐹 (𝑥, 𝑦) be an irreducible integral binary form of degree 𝑛 ⩾ 3 with
𝑠 + 1 nonzero coefficients and let ℎ be a positive integer. Then if 𝑛 > 4𝑠𝑒2Φ, the total
number of primitive solutions to |𝐹 (𝑥, 𝑦) | ⩽ ℎ satisfies
𝑁 (𝐹, ℎ) ≪ 𝑠𝑒Φℎ2/𝑛.
We will also examine the case where 𝑠 = 2 (in this case, 𝐹 (𝑥, 𝑦) is called a trinomial
because it is the sum of three nonzero terms) and improve explicit bounds on the number
of solutions to the particular Thue equations 𝐹 (𝑥, 𝑦) = ±1. The following theorem is our
main explicit result:
Theorem 1.17. Let 𝐹 (𝑥, 𝑦) = ℎ 𝑥𝑛 + ℎ 𝑥𝑘 𝑦𝑛−𝑘𝑛 𝑘 + ℎ0𝑦𝑛 where ℎ𝑛, ℎ𝑘 , ℎ0, 𝑛, 𝑘 ∈ Z with
0 < 𝑘 < 𝑛. Suppose that 𝐹 (𝑥, 𝑦) is irreducible over Z[𝑥, 𝑦] and 𝑛 ⩾ 6. Then there are at
most 2𝑣(𝑛)𝑧(𝑛) + 8 distinct integer pair solutions to the equation |𝐹 (𝑥, 𝑦) | = 1 where3 if 𝑛 is odd𝑣(𝑛) = 4 if 𝑛 is even
and 𝑧(𝑛) is defined by the following table.
𝑛 6 7 8 9 10–11 12–16 17–37 38–216 ⩾ 217
𝑧(𝑛) 15 12 11 9 8 7 6 5 4
The main method by which we are able to achieve these results is an improvement in a
counting technique associated with The Gap Principle. The Gap Principle is not a single
explicit result, but rather a series of related results that states that when two rational
20
numbers approximate the same algebraic number, their denominators must be
exponentially far apart. The counting technique improvement is Lemma 2.19. We apply
this technique to the versions of The Gap Principle which we give towards the beginning
of the proof of Lemma 2.23 and which we give as Theorem 2.42.
1.4 Root Distribution
Another feature to observe about Proposition 1.14 is that solutions to the Thue
equation 𝐹 (𝑥, 𝑦) = ℎ provide good rational approximations to one of relatively few roots
of 𝐹 (𝑥, 1) or 𝐹 (1, 𝑦). There are 2𝑛 such roots, but solutions to 𝐹 (𝑥, 𝑦) = ℎ provide good
rational approximations (in the sense of Proposition 1.14) to elements of some subset of
those roots with size at most 12𝑠 + 8. This gives us some information about how the roots
of 𝐹 (𝑥, 1) and 𝐹 (1, 𝑦) are distributed in the complex plane when 𝑠 is small relative to 𝑛:
either
1. there are few roots which are able to be well-approximated by rational numbers or
2. the roots come in clusters so that a good rational approximation of one such root is a
good approximation of the roots in the nearby cluster.
While we do not discuss the full generality needed to understand root clustering and
equidistribution, we will explore the notion of root distribution by studying how near two
roots of the same polynomial can get.
The study of polynomial root distribution originates in the broader undertaking to find
solutions to equations. Of course, in antiquity, “equations” referred to “polynomial
equations” and as we now know, “finding solutions” is often too much to ask. However, we
can glean information about the roots of a polynomial from information about the
coefficients, and that is the essence of the study of polynomial root distribution. Major
questions of this field include:
Question 1.18. For a polynomial 𝑓 (𝑥) ∈ R[𝑥], how many of its roots are real?
Question 1.19. For a polynomial 𝑓 (𝑥) ∈ C[𝑥], is there an “easily computed” compact
region of the plane in which its roots must live?
Question 1.20. For a polynomial 𝑓 (𝑥) ∈ R[𝑥], how close together can its roots be?
21
Our goal for this discussion will be to understand polynomials with integer
coefficients, but as is often the case in number theory, it is helpful to consider the fields C
and R.
1.4.1 Number of Real Roots
We can address Question 1.18 without introducing any new terminology. Let
𝑓 (𝑥) ∈ R[𝑥] be a polynomial of degree 𝑛. We begin by examining upper bounds on the
number of real roots of 𝑓 (𝑥).
Naïvely, 𝑓 (𝑥) can have no more than 𝑛 real roots because 𝑓 (𝑥) has no more than 𝑛
roots. However, this is not necessarily a good bound and in many cases, we can do better.
The following lemma is a corollary of Descartes’ rule of signs and is also given in [Sch87].
Lemma 1.21. Suppose that 𝑓 (𝑥) ∈ R[𝑥] is a polynomial with 𝑠 + 1 nonzero summands.
Then 𝑓 (𝑥) has no more than 2𝑠 + 2 real roots.
This lemma gives some indication of why Proposition 1.14 is true. After all, solutions
to the Thue equation 𝐹 (𝑥, 𝑦) = ℎ produce good rational approximations of the roots of
𝐹 (𝑥, 1) and 𝐹 (1, 𝑦). However, rational numbers can only produce arbitrarily good
approximations of real numbers, so we expect that solutions to 𝐹 (𝑥, 𝑦) = ℎ produce good
rational approximations to only the roots of 𝐹 (𝑥, 1) and 𝐹 (1, 𝑦) that lie near or on the real
axis. Lemma 1.21 indicates that the number of such roots is controlled by 𝑠, so we would
expect that solutions to 𝐹 (𝑥, 𝑦) = ℎ produce approximations to some number of roots of
𝐹 (𝑥, 1) or 𝐹 (1, 𝑦) where that number is controlled by 𝑠.
On the other hand, finding lower bounds on the number of real roots of 𝑓 (𝑥) is a much
more difficult subject. It is difficult in general to detect if 𝑓 (𝑥) has any real roots at all. Of
course, if the degree of 𝑓 (𝑥) is odd, then 𝑓 (𝑥) has real roots, but if the degree of 𝑓 (𝑥) is
even, this becomes much more difficult. However, it is possible to do this for 𝑓 (𝑥) with
rational coefficients in an effective manner.
The polynomial 𝑓 (𝑥) has a real root if and only if the∑number field 𝐾 := Q[𝑥] / ( 𝑓 (𝑥))
has an embedding into R. Letting 𝑄(𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑥 ) = 5 25 𝑖=1 𝑥 , Meyer’s Theorem𝑖
combined with the Hasse-Minkowski Theorem (see [Ser73]) combine to indicate that
there exists an embedding 𝐾 → R if and only if 𝑄 does not nontrivially represent 0 over
𝐾 . However, Raghavan in [Rag75] gives an effective algorithm for detecting whether or
not 𝑄 represents 0 nontrivially over 𝐾 and so we have an effective method for checking
whether or not 𝑓 (𝑥) has a real root.
22
1.4.2 Roots in a Compact Region
Question 1.19 is also straig∑htforward to address.
Proposition 1.22. Let 𝑓 (𝑥) = 𝑛 𝑗𝑗=0 𝑎 𝑗𝑥 ∈ C[𝑥] where 𝑎𝑛, 𝑎0 ≠ 0. If 𝛼 ≠ 0 is a root of
𝑓 (𝑥), then
1 𝐻 ( 𝑓 )
+ ( )/| | < |𝛼 | < 1 +1 𝐻 𝑓 𝑎0 |𝑎𝑛 |
.
Proof. Note that since 𝛼 is a root of 𝑓 (𝑥), we∑︁can write𝑛−1
−𝑎𝑛𝛼𝑛 = 𝑎 𝑗𝛼 𝑗 .
𝑗=0
Taking absolute valu∑︁es on both sides and∑︁applying the triangle inequality yields𝑛−1 𝑛−1 𝑛 𝑛
|𝑎𝑛 | |𝛼 |𝑛 ⩽ |𝑎 𝑗 | |𝛼 | 𝑗 ⩽ 𝐻 ( 𝑓 ) |𝛼 | 𝑗
𝐻 ( 𝑓 ) ( |𝛼 | − 1) 𝐻 ( 𝑓 ) |𝛼 |
=
0 0 | | −
< | | − .𝛼 1 𝛼 1
𝑗= 𝑗=
Dividing
| | | |𝑛 𝐻 ( 𝑓 ) |𝛼 |
𝑛
𝑎𝑛 𝛼 < |𝛼 | − 1
by |𝛼 |𝑛 and rearranging yields the desired inequality
| | + 𝐻 ( 𝑓 )𝛼 < 1 | .𝑎𝑛 |
To get the lower bound on 𝛼, observe that the reciprocal polynomial 𝑓 𝑅 (𝑥) = 𝑥𝑛 𝑓 (1/𝑥)
has height 𝐻 ( 𝑓 𝑅) = 𝐻 ( 𝑓 ) and has 1/𝛼 as a root. By the first part of the proof then,
1 𝐻 ( 𝑓 𝑅)
| < 1 +𝛼 | |𝑎0 |
and taking reciprocals yields the desired lower bound on |𝛼 |. □
Moreover, it is not difficult to see that these are good bounds on the roots of 𝑓 (𝑥). For
example, for 𝑡 ⩾ 1, the family of polynomials 𝑓𝑡 (𝑥) = (𝑥𝑛−1 − 1) (𝑥 − 𝑡) has height 𝑡,
leading coefficient 1, and a root located at 𝑥 = 𝑡. Proposition 1.22 indicates that the roots
of 𝑓𝑡 (𝑥) must live in the circle with radius 1 + 𝑡 centered at 0. As 𝑡 tends to infinity then,
the ratio of the size of the largest root of 𝑓𝑡 (𝑥) to the upper bound on root size given by
Proposition 1.19 tends to 1, so the bound is sharp. A similar construction can be given to
show that the lower bound is sharp.
23
1.4.3 Separation of Polynomial Roots
Question 1.20 is trickier to address and we give an overview to this question here
before covering it in more depth in chapter 3. We want to understand the question more
precisely, however, so we will introduce the following quantities.
Definition 1.23. Given a polynomial 𝑓 (𝑥1, . . . , 𝑥𝑚) ∈ C[𝑥1, . . . , 𝑥𝑚], the Mahler measure
of 𝑓 (𝑥1, . . . , 𝑥𝑚) is the(∫quan∫tity1 1 ∫ 1 )
𝑀 ( 𝑓 ) = exp · · · log | 𝑓 (𝑒2𝜋𝑖𝑡1 , . . . , 𝑒2𝜋𝑖𝑡𝑚) | 𝑑𝑡1 𝑑𝑡2 · · · 𝑑𝑡𝑚 .
0 0 0
Definition 1.24. Given a polynomial 𝑓 (𝑥) ∈ C[𝑥] of degree 𝑛, with roots 𝛼1, . . . , 𝛼𝑛 ∈ C,
and with leading coefficient 𝑏, the discrimin∏ant of 𝑓 (𝑥) is the quantity
Δ = 𝑏2𝑛−2𝑓 (𝛼𝑖 − 𝛼 𝑗 )2.
1⩽𝑖< 𝑗⩽𝑛
Definition 1.25. Given a polynomial 𝑓 (𝑥) ∈ C[𝑥] with roots 𝛼1, . . . , 𝛼𝑛 ∈ C, the
separation of 𝑓 (𝑥) is the quantity
sep( 𝑓 ) = min |𝛼𝑖 − 𝛼 𝑗 |.
𝛼𝑖≠𝛼 𝑗
Before giving any answers to Question 1.20, we will address some of the connections
between the roots, the height, the Mahler measure, and the discriminant of a polynomial
𝑓 (𝑥) ∈ C[𝑥].
Lemma 1.26. For a single-variable polynomial 𝑓 (𝑥) ∈ C[𝑥] with roots 𝛼1, . . . , 𝛼𝑛 ∈ C
and leading coefficient 𝑏, we have ∏𝑛
𝑀 ( 𝑓 ) = |𝑏 | max(1, |𝛼 𝑗 |).
𝑗=1
This is Proposition 1.6.5 in [BG06]. This result is a corollary of the more general
Jensen’s formula (see [Rud74]) and it gives an important connection between the Mahler
measure of a polynomial and the polynomial’s roots. Moreover, the Mahler measure
satisfies the following dual inequalities, stated as Lemma 1.6.7 in [BG06]:
Lemma 1.27. Su(ppose t)hat 𝑓 (𝑥) ∈ C[𝑥] has degree 𝑛 and 𝑠 + 1 nonzero summands. Then−1
𝑛 √ √
⌊ 𝐻 ( 𝑓 ) ⩽ 𝑀 ( 𝑓 ) ⩽ 𝐻 ( 𝑓 ) 𝑠 + 1 ⩽ 𝐻 ( 𝑓 ) 𝑛 + 1. (1.6)𝑛/2⌋
24
This lemma shows that, up to a constant factor depending on the degree of 𝑓 (𝑥), the
Mahler measure and height of 𝑓 (𝑥) are the same. This is what makes the Mahler measure
a key quantity in number theory: it connects the known coefficients to the unknown roots.
The discriminant of 𝑓 (𝑥) is important for the same reason. Its definition is given in
terms of the roots of 𝑓 (𝑥), but it has a key connection to the coefficients of 𝑓 (𝑥) as well.
Note that the definition of the discriminant given in 1.24 has Δ 2𝑛−2𝑓 /𝑏 given as a
polynomial in 𝛼1, . . . , 𝛼𝑛. Moreover, Δ 𝑓 /𝑏2𝑛−2 is a symmetric polynomial in 𝛼1, . . . , 𝛼𝑛
and hence, by Milne’s proof of the Fundamental Theorem of Symmetric Polynomials (see
Theorem 2.2 in [Mil20]), Δ /𝑏2𝑛−2𝑓 can be expressed as a polynomial of degree equal to
2𝑛 − 2 in the elementary symmetric polynomials, 𝑒1(𝛼1, . . . , 𝛼𝑛), . . . , 𝑒𝑛 (𝛼1, . . . , 𝛼𝑛).
However, if the leading coefficient of 𝑓 (𝑥) is 𝑏, then the coefficients of 𝑓 (𝑥) are equal to
(up to sign) 𝑏𝑒1(𝛼1, . . . , 𝛼𝑛), . . . , 𝑏𝑒𝑛 (𝛼1, . . . , 𝛼𝑛). Letting 𝑏𝑖 denote 𝑏𝑒𝑖 (𝛼1, . . . , 𝛼𝑛), we
find that Δ /𝑏2𝑛−2𝑓 can be expressed as a polynomial of degree 2𝑛 − 2 in the variables
𝑏𝑖/𝑏. Hence, Δ 𝑓 can be expressed as a polynomial of degree 2𝑛 − 2 in the variables 𝑏𝑖, i.e.
the coefficients of 𝑓 (𝑥).
While it is helpful to know that the discriminant can be defined in terms of the roots or
in terms of the coefficients of 𝑓 (𝑥), the previous paragraph gives no indication of how the
discriminant can be expressed in terms of the coefficients. A tool known as the resultant
helps with that particular con∑cern: ∑
Definition 1.28. Let 𝑓 (𝑥) = 𝑛 𝑖𝑖=0 𝑎𝑖𝑥 and 𝑔(𝑥) =
𝑚 𝑗
𝑗=0 𝑏 𝑗𝑥 be polynomials with complex
coefficients with 𝑎𝑛, 𝑏𝑚 ≠ 0. Then the resultant of 𝑓 (𝑥) and 𝑔(𝑥) is the determinant of the
(𝑛 + 𝑚) × (𝑛 + 𝑚) Sylvester matrix
­­© 𝑎𝑛 0 · · · 0 𝑏𝑚 0 · · · 0­ ª­ . . . . ®­𝑎𝑛−1 𝑎 .𝑛 . .. 𝑏𝑚−1 𝑏 .𝑚 . .. ®­ ®­ . . 0 . ®­𝑎𝑛−2 𝑎 .𝑛−1 . 𝑏𝑚−2 𝑏𝑚−1 . 0 ®­­ .
®
.. .
. .
. . . 𝑎 .
. . . ®
( ) ­ 𝑛 .
.. . . 𝑏
Res 𝑚®𝑓 , 𝑔 := det ­­ . .
® .
𝑎0 𝑎1 · · · .. 𝑏0 𝑏1 · · · .. ®
­­
®®
­ 0 .𝑎0 . .. .. 0 .𝑏0 . ... . ®­ ®.. . . . . . ®« . .. . . 𝑎1 .. .. . . 𝑏1 ®®0 0 · · · 𝑎0 0 0 · · · 𝑏0 ¬
It turns out that the discriminant of a polynomial 𝑓 (𝑥) is closely related to the
resultant of 𝑓 (𝑥) and its derivative 𝑓 ′(𝑥). The following lemma can be found after the
proof of Proposition 2.34 in [Mil20].
25
Lemma 1.29. If 𝑓 (𝑥) ∈ C[𝑥] is a polynomial with degree 𝑛 and leading coefficient 𝑏 ≠ 0,
then ′
Δ = (−1)𝑛(𝑛−1)/2 · Res( 𝑓 , 𝑓 )𝑓 .
𝑏
This gives a straightforward way to compute the discriminant as a polynomial in the
coefficients of 𝑓 (𝑥). Before we begin to address Question 1.20, we will look at an
important class of examples.
Example 1.30. Consider the family of polynomials 𝑄𝑛,𝑟 (𝑥) = 𝑥𝑛 − 𝑟 for real 𝑟 ⩾ 1 and
integer 𝑛 ⩾ 2. Let 𝜁𝑛 denote a primitive 𝑛th root of unity. Then we have:
𝐻 (𝑄𝑛,𝑟) =∏max(1, 𝑟) = 𝑟𝑛
𝑀 (𝑄𝑛,𝑟) = max(1, | 𝑗𝜁𝑛𝑟1/𝑛 |) = 𝑟
𝑗=1
Δ = (−1)𝑛−1𝑛𝑛𝑟𝑛−1𝑄𝑛,𝑟
sep(𝑄 1/𝑛𝑛,𝑟) = 𝑟 √︃|𝑒2𝜋𝑖/𝑛 − 1|
= 𝑟1/𝑛√︁ (1 − cos(2𝜋/𝑛))2 + sin2(2𝜋/𝑛)
= 𝑟1/𝑛√︃2 − 2 cos(2𝜋/𝑛)
= 𝑟1/𝑛 2 − 2(cos2(𝜋/𝑛) − sin2(𝜋/𝑛))
= 2𝑟1/𝑛 sin(𝜋/𝑛)
Importantly, note that for this class of examples,
𝐻 (𝑄𝑛,𝑟 = 𝑀 (𝑄𝑛,𝑟)
and (𝜋 )
sep(𝑄 1/𝑛𝑛,𝑟) = 2 sin · 𝑀 (𝑄𝑛,𝑟) .
𝑛
A key relation which helps to address Question 1.20 was shown by Mahler in [Mah64]:
Theorem 1.31 (Mahler). Let 𝑓 (𝑥) ∈ C[𝑥] hav√︁e degree 𝑛 ⩾ 2. Then
3|Δ |
sep( ) 𝑓𝑓 > .
𝑛(𝑛+2)/2𝑀 ( 𝑓 )𝑛−1
In particular, if we suppose that 𝑓 (𝑥) is separable, then |Δ 𝑓 | > 0 by Definition 1.24.
Furthermore, |Δ 𝑓 | is a polynomial in the coefficients of 𝑓 (𝑥). So if 𝑓 (𝑥) ∈ Z[𝑥]—the
primary case we are interested in—then |Δ 𝑓 | must be a positive integer. Hence, we must
have |Δ 𝑓 | ⩾ 1. As a result, we have the following corollary.
26
Corollary 1.32 (Mahler). Suppose that 𝑓 (𝑥) ∈ Z[𝑥] is separable of degree 𝑛 ⩾ 2. Then
√
3
sep( 𝑓 ) > .
𝑛(𝑛+2)/2𝑀 ( 𝑓 )𝑛−1
This corollary is the standard to which other theorems on polynomial root separation
are often compared. Philosophically, it indicates that the roots of polynomials with integer
coefficients repel each other to some extent.
1.4.4 New Results on Polynomial Root Separation
In Chapter 3, we reverse the question that Mahler answers with Theorem 1.31 and we
ask about upper bounds for sep( 𝑓 ) in terms of 𝑀 ( 𝑓 ).
Naïvely, Lemma 1.22 gives an upper bound on separation in terms of the height of the
polynomial, which can then be translated into a bound on separation in terms of the
Mahler measure. Lemma 1.22 indicates that any two roots of 𝑓 (𝑥) ∈ C[𝑥] of degree 𝑛
must satisfy |𝛼 − 𝛽 | < 2 + 2𝐻 ( 𝑓 ) and by Lem(ma 1.2)7, we find that
𝑛
sep( 𝑓 ) < 2 + 2 ⌊ / 𝑀 ( 𝑓 ). (1.7)𝑛 2⌋
However, this is a crude estimate and can be dramatically improved, as we will show in
Proposition 3.4. We conjecture that this estimate can be improved in the following way:
Conjecture 1.33. Suppose 𝑓 (𝑥) ∈ R[𝑥] is monic and separable of degree 𝑛 ⩾ 2. If 𝑓 (𝑥)
has any real roots, then
sep( 𝑓 ) ≪𝑛 𝑀 ( 𝑓 )1/(𝑛−1) .
If 𝑓 (𝑥) has only nonreal roots, then
sep( 𝑓 ) ≪ 1/𝑛𝑛 𝑀 ( 𝑓 ) .
We support this conjecture with data, families of examples, and the following theorem.
Theorem 1.34. Let 𝑓 (𝑥) ∈ R[𝑥] be monic and separable with deg( 𝑓 ) = 𝑛 ⩾ 2 and
suppose that any of the following conditions is met.
1. deg( 𝑓 ) = 2.
2. deg( 𝑓 ) = 3.
3. deg( 𝑓 ) = 4 and 𝑓 (𝑥) has no real roots.
27
4. Every root of 𝑓 (𝑥) is real.
Then if 𝑓 (𝑥) has any real roots,
sep( 𝑓 ) ≪ 𝑀 ( 𝑓 )1/(𝑛−1)𝑛 .
If 𝑓 (𝑥) has only nonreal roots, then
sep( 𝑓 ) ≪ 𝑀 ( 𝑓 )1/𝑛𝑛 .
We prove this theorem in pieces and we consider each condition separately. We mainly
prove this by analyzing the geometry of possible root locations in the complex plane.
28
CHAPTER 2
THUE EQUATIONS
2.1 Introduction
Axel Thue, in [Thu09], showed that when ℎ is an integer and when 𝐹 (𝑥, 𝑦) is an
irreducible integral binary form (recall Definition 1.10) with degree 𝑛 ⩾ 3, the equation
𝐹 (𝑥, 𝑦) = ℎ
has only finitely many integer-pair solutions. We have previously stated this equation as
equation (1.3) and we will continue to use this number to refer to it during this chapter.
Recall that we will only use the word “solution” to refer to integer-pair solutions.
It is worth noting first that the each of the hypotheses is necessary. The famous Pell
equation 𝑥2 − 𝑑𝑦2 = 1 has infinitely many integer solutions and it meets every hypothesis
except for deg(𝐹) ⩾ 3. If 𝐹 (𝑥, 𝑦) is not required to be irreducible, then it may have a
linear factor, say 𝑚𝑥 − 𝑛𝑦. But then any multiple of the pair (𝑛, 𝑚) will satisfy 𝐹 (𝑥, 𝑦) = 0
and this will give infinitely many solutions to 𝐹 (𝑥, 𝑦) = 0. If 𝐹 (𝑥, 𝑦) is not required to be
homogeneous, then again we can acquire infinitely many solutions to 𝐹 (𝑥, 𝑦) = 0 as, for
example, the equation 𝑥6 + 𝑦3 = 0 has infinitely many solutions of the form (𝑛,−𝑛2).
As a consequence of Thue’s result, the inequality
|𝐹 (𝑥, 𝑦) | ⩽ ℎ (2.1)
also has finitely many integer-pair solutions. This inequality—called Thue’s
Inequality—can often be easier to work with because it treats the solutions to the equations
|𝐹 (𝑥, 𝑦) | = 1, . . . , |𝐹 (𝑥, 𝑦) | = ℎ
in aggregate so we deal primarily with Thue’s inequality in this paper.
Several natural questions arise from Thue’s results such as
1. How many solutions are there to (2.1)?
2. How large are solutions to (2.1)?
3. On which features of 𝐹 and ℎ do the solutions to (2.1) depend?
This chapter largely handles the first question, though of course the second and third
questions are related. In particular, the number of nonzero summands of 𝐹 (𝑥, 𝑦)
significantly impacts the number of solutions to (2.1).
29
The rough reasoning for this is as follows: a solution (𝑝, 𝑞) to (2.1) gives a good
rational approximation 𝑝/𝑞 to a root of 𝑓 (𝑋) := 𝐹 (𝑋, 1) as we saw previously in
Proposition 1.13. The only roots of 𝑓 (𝑋) which ought to allow good rational
approximations are the real roots of 𝑓 (𝑋). It is the number of nonzero coefficients of
𝑓 (𝑋) that controls the number of real roots of 𝑓 (𝑋), as we previously saw in Lemma 1.21,
so we expect the number of nonzero summands of 𝑓 (𝑋) to play a role in bounding the
number of solutions to (2.1).
Moreover, this connection between solutions to (2.1) and rational approximations to
roots of 𝑓 (𝑋) gives us reason to initially count only the solutions (𝑝, 𝑞) with
gcd(𝑝, 𝑞) = 1: pairs (𝑝, 𝑞) with gcd(𝑝, 𝑞) = 1 and 𝑞 > 0 are in bijection with rational
numbers 𝑝/𝑞. This motivates the following definition.
Definition 2.1. A pair (𝑝, 𝑞) ∈ Z2 is said to be primitive if gcd(𝑝, 𝑞) = 1.
Once we have a bound on the number of primitive solutions to (2.1), we can often use
these to bound the total number of solutions to (2.1) with partial summation techniques
(see the discussion after the statement of Proposition 3 in [MS88], for instance).
Our first theorem regards asymptotic bounds on the number of primitive solutions to
(2.1), so take a moment to review the definition of the≪ symbol from Notation 1.15.
Additionally, we introduce the following piece of notation:
Notation 2.2. Let 𝑁 (𝐹, ℎ) denote the total number of primitive integer-pair solutions to
(2.1).
Mueller and Schmidt in [MS88] prove that
Theorem 2.3 (Mueller and Schmidt). Let 𝐹 (𝑥, 𝑦) be an irreducible integral binary form
of degree 𝑛 ⩾ 3 with 𝑠 + 1 nonzero coefficients and let ℎ be a positive integer. The number
of primitive integer solutions of the inequality |𝐹 (𝑥, 𝑦) | ⩽ ℎ satisfies
𝑁 (𝐹, ℎ) ≪ 𝑠2ℎ2/𝑛 (1 + log ℎ1/𝑛). (2.2)
Moreover, if 𝑛 ⩾ 𝑠 log3 𝑠, then
𝑁 (𝐹, ℎ) ≪ 𝑠2ℎ2/𝑛.
In the same paper, Mueller and Schmidt conjecture that the exponent on 𝑠 can be
improved. We later explain the heuristics behind this conjecture and state their conjecture
as Conjecture 2.5. Our theorem (which is stated first as Theorem 1.16, but reprinted here
for convenience) improves the exponent on 𝑠 at the cost of a stronger assumption on 𝑛.
30
The parameter Φ will be defined at the beginning of Section 2.3 and it satisfies
log3(𝑠) ≪ 𝑒Φ ≪ 𝑠.
Theorem. Let 𝐹 (𝑥, 𝑦) be an irreducible integral binary form of degree 𝑛 ⩾ 3 with 𝑠 + 1
nonzero coefficients and let ℎ be a positive integer. Then if 𝑛 > 4𝑠𝑒2Φ, the total number of
primitive solutions to |𝐹 (𝑥, 𝑦) | ⩽ ℎ satisfies
𝑁 (𝐹, ℎ) ≪ 𝑠𝑒Φℎ2/𝑛. (2.3)
Since the number of nonzero summands plays such an important role, we have names
for polynomials which meet specific values of 𝑠.
Definition 2.4. If a polynomial has exactly two nonzero summands, it is called a binomial;
if it has exactly three nonzero summands, it is called a trinomial; and if it has exactly four
nonzero summands, it is called a tetranomial.
Authors such as Bennett [Ben01], Evertse [Eve82], Grundman and Wisniewski
[GW13], Hyyrö [Hyy64], Mueller [Mue87], Mueller and Schmidt [MS87], and Thomas
[Tho00] have examined binomial, trinomial, and tetranomial Thue equations in hopes to
get a better handle on how the number of nonzero summands of 𝐹 (𝑥, 𝑦) affect the number
of solutions to Thue equations.
In particular, Bennett’s result that there is at most one solution in positive integers to
𝐹 (𝑥, 𝑦) = 1 for binomial 𝐹 (𝑥, 𝑦) is worth mentioning because that is the best possible
result: the infinite family of binomial Thue equations
(𝑎 + 1)𝑥𝑛 − 𝑎𝑦𝑛 = 1
always has a solution at 𝑥 = 1, 𝑦 = 1.
In this chapter, we will also improve explicit bounds for the number of solutions to the
Thue equation
|𝐹 (𝑥, 𝑦) | = 1 (2.4)
in the particular case that 𝐹 (𝑥, 𝑦) is a trinomial. The case where ℎ = ±1 is foundational to
the study of Thue equations as many bounds for general ℎ can be derived from knowing
the number of solutions to |𝐹 (𝑥, 𝑦) | = 1. We will discuss this in more detail at the
beginning of Section 2.2.
31
In the setting where 𝐹 (𝑥, 𝑦) is a trinomial, Thomas showed in [Tho00] that there are
no more than 2𝑣(𝑛)𝑤(𝑛) + 8 distinct integer pair solutions to |𝐹 (𝑥, 𝑦) | = 1 when
𝐹 (𝑥, 𝑦) ∈ Z[𝑥, 𝑦] is a trinomial irreducible binary form of degree 𝑛 ⩾ 3 where3 if 𝑛 is odd𝑣(𝑛) = 4 if 𝑛 is even
and 𝑤(𝑛) is defined by the following table.
𝑛 51 6 7 8 9 10–11 12–16 17–37 ⩾ 38
𝑤(𝑛) 271 16 13 11 9 8 7 6 5
1 There is an error in the proof of Lemma 4.1 in [Tho00]: it is
claimed that 𝑏𝑡−1 𝑡−1 < 𝑏 which is not the case for the choice of𝑏
𝑏 = 1.5when 𝑛 = 5. Tracing this error through to its conclusion,
the author believes that this is not correctable and that Thomas’
work does not yield a result when 𝑛 = 5.
We are able to improve the bounds that Thomas provides and we have the following
theorem (stated first as Theorem 1.17, but reprinted here for convenience).
Theorem. Let 𝐹 (𝑥, 𝑦) = ℎ 𝑛 𝑘 𝑛−𝑘 𝑛𝑛𝑥 + ℎ𝑘𝑥 𝑦 + ℎ0𝑦 where ℎ𝑛, ℎ𝑘 , ℎ0, 𝑛, 𝑘 ∈ Z with
0 < 𝑘 < 𝑛. Suppose that 𝐹 (𝑥, 𝑦) is irreducible over Z[𝑥, 𝑦] and 𝑛 ⩾ 6. Then there are at
most 2𝑣(𝑛)𝑧(𝑛) + 8 distinct integer pair solutions to the equation |𝐹 (𝑥, 𝑦) | = 1 where3 if 𝑛 is odd𝑣(𝑛) = 4 if 𝑛 is even
and 𝑧(𝑛) is defined by the following table.
𝑛 6 7 8 9 10–11 12–16 17–37 38–216 ⩾ 217
𝑧(𝑛) 15 12 11 9 8 7 6 5 4
Both of our main results are primarily derived from an improvement in efficiency to a
counting technique associated with the gap principle (see Lemma 2.19).
2.2 Conjectures and Heuristics
Baker in [Bak67] showed that integer solutions to (2.1) all satisfy
max( |𝑥 |, |𝑦 |) < 𝐶𝑒(log ℎ)𝜅 where 𝜅 is any real number larger than the degree of 𝐹 and 𝐶 is
an effectively computable constant depending only on 𝐹 and 𝜅. This result provides some
32
insight to the number of solutions to (2.1) in that it gives an upper bound for the number of
solutions. Explicitly, the total number of integer pairs (𝑥, 𝑦) satisfying
𝜅
max( |𝑥 |, |𝑦 |) < 𝐶𝑒(log ℎ)
is
(2⌊𝐶𝑒(log ℎ)𝜅 ⌋ + 1)2.
Then the number of integer pair solutions to (2.1) must also be no larger than
(2⌊ 𝜅𝐶𝑒(log ℎ) ⌋ + 1)2,
though this upper bound is not sharp.
In particular, we can see that the dependence of the upper bound on ℎ can be much
improved by considering the geometry of the situation. The set
𝑆(𝐹, ℎ) := {(𝑥, 𝑦) ∈ R2 : |𝐹 (𝑥, 𝑦) | ⩽ ℎ}
is a star body of finite volume (the latter claim is true, though non-obvious) and intuition
from the geometry of numbers indicates that the number of integer pair solutions to
(2.1)—which correspond exactly to the integer lattice points inside 𝑆(𝐹, ℎ)—should be
approximately equal to vol(𝑆(𝐹, ℎ)). But if 𝐹 has degree 𝑛, then
vol(𝑆(𝐹, ℎ)) = vol{{(𝑥, 𝑦) ∈ R2 : |𝐹 ((𝑥, 𝑦) | ⩽ ℎ} )𝑥 𝑦  }
= vol (𝑥, 𝑦) ∈ R2 : 𝐹 , 1/ ⩽ 1ℎ 𝑛 ℎ1/𝑛 
= vol{(ℎ1/𝑛𝑥, ℎ1/𝑛𝑦) ∈ R2 : |𝐹 (𝑥, 𝑦) | ⩽ 1}
= ℎ2/𝑛 vol(𝑆(𝐹, 1)).
Based on this fact, we might guess that the number of solutions to to (2.1) is bounded
above by a constant depending on 𝐹 times ℎ2/𝑛. This turns out to be the case as Mahler in
[Mah33] showed that the number of solutions to (2.1) is equal to
vol(𝑆(𝐹, 1))ℎ2/𝑛 +𝑂 (ℎ1/(𝑛−1)𝐹 ).
Since this essentially gives the dependence on ℎ of the number of solutions to (2.1)
when ℎ is large, it more or less remains to examine how many solutions there are to (2.4)
and multiply that result by ℎ2/𝑛.
From here, note that we have already observed in our discussion after Lemma 1.21
that solutions to (2.4) yield good rational approximations of relatively few roots of 𝐹 (𝑥, 1)
33
and 𝐹 (1, 𝑦). Specifically, solutions yield good rational approximations to one of no more
than 8𝑠 + 12 algebraic numbers in the sense of Proposition 1.14. If there are boundedly
many good rational approximations per root (as seems plausible, though we have not
argued for this) and the number of roots which solutions can approximate is no more than
a constant times 𝑠, then we would expect that the number of solutions to (2.4) would be no
more than a constant times 𝑠. Adding in dependence on ℎ, we arrive at the following
conjecture which Mueller and Schmidt made in [MS88].
Conjecture 2.5 (Mueller and Schmidt). Let 𝐹 (𝑥, 𝑦) be an irreducible integral binary form
of degree 𝑛 ⩾ 3 with 𝑠 + 1 nonzero coefficients and let ℎ be a positive integer. The number
of primitive solutions of |𝐹 (𝑥, 𝑦) | ⩽ ℎ satisfies
𝑁 (𝐹, ℎ) ≪ 𝑠ℎ2/𝑛.
This remains a conjecture at the moment, though there are quite a few results moving
in the direction of this bound. Initial work was done by Schmidt in [Sch87] to show that
the total number of solutions to (2.1) satisfies
𝑁 (𝐹, ℎ) ≪ (𝑛𝑠)1/2ℎ2/𝑛 (1 + log ℎ2/𝑛). (2.5)
Mueller and Schmidt later eliminated the dependence of the bound on the degree 𝑛
and replaced it by a more suitable dependence on 𝑠. This is where Theorem 2.3, which is
from [MS88], enters the picture.
Thunder improves the logarithmic factor of (2.2) in [Thu95] when ℎ is large relative to
the discriminant of 𝐹 and Akhtari and Bengoechea in [AB20] have improved both the
exponent on 𝑠 and the logarithmic factor when ℎ is small relative to the discriminant of 𝐹.
Saradha and Sharma in [SS17] improve the exponent on 𝑠 without assuming any
restrictions on ℎ.
When ℎ is small compared to the coefficients of 𝐹, one might expect that neither ℎ nor
the specific coefficients of 𝐹 play a serious role in the number of solutions to (2.1). To that
end, letting 𝐻 (𝐹) denote the height of 𝐹 (𝑥, 𝑦) (recall Definition 1.12), Mueller and
Schmidt in [MS88] gave the following conjecture:
Conjecture 2.6 (Mueller and Schmidt). Let 𝐹 (𝑥, 𝑦) be an irreducible integral binary form
of degree 𝑛 ⩾ 3 with 𝑠 + 1 nonzero coefficients and let ℎ be a positive integer. If 𝜌 > 0 and
𝑠
ℎ ⩽ 𝐻 (𝐹)1− −𝜌𝑛 , then the number of primitive solutions to |𝐹 (𝑥, 𝑦) | ⩽ ℎ is less than or
equal to a function of only 𝑠 and 𝜌.
34
Mueller in [Mue87] and Mueller and Schmidt in [MS87] have confirmed Conjecture
2.6 in the cases of 𝑠 = 1, 2 while Thomas in [Tho00], and Grundman and Wisniewski in
[GW13] provide evidence for this conjecture in the cases of 𝑠 = 2, 3, but the conjecture
remains open otherwise. Techniques in these cases are varied and often rely on the
improved approximation results one can acquire by fixing a small value of 𝑠.
Alternatively, one could compare the size of ℎ to the discriminant of 𝐹 and in that
case, Akhtari and Bengoechea in [AB20] show that
Theorem 2.7 (Akhtari and Bengoechea). Let 𝐹 (𝑥, 𝑦) be an irreducible integral binary
form of degree 𝑛 ⩾ 3 with 𝑠 + 1 nonzero coefficients and let ℎ be a positive integer. If 𝐹
has discriminant Δ𝐹 and
1
|Δ | 8(𝑛−1)
0 𝐹< ℎ < ,
(3 2𝑛800 log 𝑛)𝑛/2(𝑛𝑠)2𝑠+𝑛
then the number of primitive solutions to the ine(quality |𝐹 (𝑥, 𝑦) |)⩽ ℎ satisfies
𝑁 ( 1𝐹, ℎ) ≪ 𝑠 log 𝑠min 1, .
log 𝑛 − log 𝑠
2.3 Notation and Definitions
Throughout the chapter,∑we use the following notation.
Suppose that 𝐹 (𝑥, 𝑦) = 𝑠𝑖=0 𝑎 𝑛𝑖 𝑛−𝑛𝑖𝑖𝑥 𝑦 ∈ Z[𝑥, 𝑦] is an irreducible binary form of
degree 𝑛 ⩾ 3 with each 𝑎𝑖 ≠ 0 so that 𝐹 (𝑥, 𝑦) has exactly 𝑠 + 1 nonzero coefficients. Set
𝐻 = 𝐻 (𝐹) to be the height of 𝐹 and let ℎ be a positive integer. Following Akhtari and
Bengoechea in [AB20], define
2 3
𝑅 = 𝑛800 log √︁𝑛 = 𝑒800 log 𝑛,
𝐶 = 𝑅ℎ(2𝐻 𝑛(𝑛 + 1))𝑛.
Following Saradha and Sharma in [SS(1∑︁7], define )𝑖−1 1 ∑︁𝑠 1
Ψ = max max , ,
0⩽𝑖⩽𝑠
0 𝑛𝑖 − 𝑛= 𝑤 = +1 𝑛𝑤 − 𝑛𝑖𝑤 𝑤 𝑖
Φ = max(Ψ, 3 log log 𝑠).
Saradha and Sharma in [SS17] note that 𝑒Ψ ≪ 𝑠, so that log3 𝑠 ≪ 𝑒Φ ≪ 𝑠.
35
Again following√A√khtari and Bengoechea in [AB20], select constants 𝑎 and 𝑏 so that
0 < 𝑎 < 𝑏 < 1 and 2 3+𝑎21− < 3√︁. Then set𝑏
2(𝑛 + 𝑎2)
𝜆 = ( − ,1 𝑏 ) 1
𝑌𝑆 = (12
𝑛−2𝑠
𝑒Ψ)𝑛𝑅2𝑠ℎ ,
1 √
𝑌 = (2𝐶) − (4𝐻 𝑛 + 1𝑒𝑛/2)𝜆/((𝑛−𝜆)𝑎2)𝐿 𝑛 𝜆 ,
( )2( Ψ)𝑛/𝑠 1/𝑠 1− 1𝐾 = 2𝑅 𝑛𝑠 12𝑒 ℎ 𝐻 𝑛 𝑠 .
For a pair x = (𝑥, 𝑦) ∈ Z2, define
⟨x⟩ = min( |𝑥 |, |𝑦 |),
|x| = max( |𝑥 |, |𝑦 |).
Then we make the following definitions.
Definition 2.8. A pair x ∈ Z2 is small if ⟨x⟩ ⩽ 𝑌𝑆, medium if 𝑌𝑆 ⩽ ⟨x⟩ and |x| < 𝑌𝐿 , and
large if |x| ⩾ 𝑌𝐿 .
To count the solutions of each type, we use the following notation.
Notation 2.9. Let 𝑁𝐿 (𝐹, ℎ) denote the number of primitive large solutions to (2.1). Let
𝑁𝑀 (𝐹, ℎ) denote the number of primitive medium solutions to (2.1). Let 𝑁𝑆 (𝐹, ℎ) denote
the number of primitive small solutions to (2.1).
Observe that the terms “small,” “medium,” and “large” are all dependent on 𝐹 (𝑥, 𝑦)
and ℎ and moreover, they are not necessarily disjoint categories.
The essential strategy for counting solutions to (2.1) is to find bounds separately for
the numbers of primitive small, medium, and large solutions. Even though there is some
overlap in the classification of large, medium, and small solutions (and hence, some
overcounting of the number of solutions), the existing techniques for counting the different
types of solutions are so disparate that it is difficult to count the overlap in any meaningful
way.
2.4 Previous Results
Describing the best existing results is difficult because of the issue of “moving
goalposts.” Authors often try to prove something along the lines of “the number of
36
solutions to (2.1) is bounded by 𝑓 (𝑛, 𝑠, ℎ)” for an appropriate function 𝑓 (𝑛, 𝑠, ℎ). To do
this, they come up with an appropriate function 𝑔(𝑠) and note something like “if 𝑛 ⩽ 𝑔(𝑠),
then (2.5) immediately implies that the number of solutions is bounded by 𝑓 (𝑛, 𝑠, ℎ).”
Then the author will proceed to count small, medium, and large solutions under the
assumption that 𝑛 > 𝑔(𝑠). Hence, the “best counts” for small, medium, and large solutions
often depend on the author’s intended upper bound, 𝑓 (𝑛, 𝑠, ℎ) and the best counts often
have the form “if 𝑛 > 𝑔(𝑠), then the number of small/medium/large solutions is bounded
𝑓 (𝑛, 𝑠, ℎ).” In particular, if we wish to prove something like Conjecture 2.5, then we first
must eliminate the logarithmic factor from (2.5) and even then, we can only use (2.5) to
show Conjecture 2.5 under the assumption that 𝑛 ≪ 𝑠.
What follows is a necessarily incomplete list of some of the best existing bounds for
counts of (primitive) large, medium, and small solutions, where we make sure to be clear
about what assumptions the authors use when it comes to the size of 𝑛 relative to 𝑠. Recall
that 𝑁𝐿 (𝐹, ℎ), 𝑁𝑀 (𝐹, ℎ), and 𝑁𝑆 (𝐹, ℎ) denote the number of primitive large, medium,
and small solutions to (2.1) respectively. For each type of solution, we make sure to
include a result where the author merely assumes 𝑛 ≫ 𝑠.
2.4.1 Large Solutions
Mueller and Schmidt in [MS88] show that
Lemma 2.10 (Mueller and Schmidt). For all 𝑛,
𝑁𝐿 (𝐹, ℎ) ≪ 𝑠.
Of the three types of solutions, this is the smallest upper bound and also the one most
closely aligned with Conjectures 2.5 and 2.6, so it has received little more attention than
what is stated here.
2.4.2 Medium Solutions
Turning our attention to medium solutions, Saradha and Sharma in [SS17] show that
Lemma 2.11 (Saradha and Sharma). Whe(n 𝑛 > 4𝑠𝑒2Φ,𝑠 ( ))
𝑁 (𝐹, ℎ) ≪ log 𝑠 + log 1 + ℎ1/𝑛𝑀 .Φ
Alternatively, Bengoechea in [Ben22] shows that
37
Lemma 2.12 (Bengoechea). When 𝑛 ⩾ 3𝑠,
 ( )1 + log ℎ1/𝑛𝑠 4 lo(g
if 𝑛 ⩾ 𝑠
𝐻
𝑁𝑀 (𝐹, ℎ) ≪ (𝑠 log 𝑠) (
log ℎ1/
)
𝑛
1 + 2log ) if 9𝑠 ⩽ 𝑛 < 𝑠4 .𝐻
( log ) 1 + 𝑠+log ℎ
1/𝑛
𝑠 𝑠 log if 𝑛 < 9𝑠
2
𝐻
These two counts of medium solutions are hard to compare since
log log 𝑠 ≪ Φ ≪ log 𝑠,
so Saradha and Sharma’s result is better when 4𝑠𝑒2Φ < 𝑛 < 𝑠4 and Bengoechea’s is better
otherwise. Additionally, Bengoechea’s result also incorporates the height 𝐻, which is
advantageous toward proving Conjecture 2.6.
Akhtari and Bengoechea in [AB20] consider the special case when ℎ is small relative
to the discriminant of 𝐹. They show the following lemma.
Lemma 2.13 (Akhtari and Bengoechea). If 𝐹 has discriminant Δ𝐹 and
1
|Δ𝐹 | 8(𝑛−1)0 < ℎ < ,
(3𝑛800 log2 𝑛)𝑛/2(𝑛𝑠)2𝑠+𝑛
then ( )
𝑁𝑀 (
1
𝐹, ℎ) ≪ 𝑠 log 𝑠min 1, .
log 𝑛 − log 𝑠
2.4.3 Small Solutions
Next, we examine results for small solutions. Saradha and Sharma in [SS17] show
Lemma 2.14 (Saradha and Sharma). There exist constants 𝑐7 and 𝑐8 so that if
𝑛 > 4𝑠𝑒2Φ,
then
𝑁 𝑐7 (log 𝑠)𝑒
−2Φ 2 Φ+𝑐8(log3 𝑠)𝑒−Φ 1
𝑆 (𝐹, ℎ) ≪ 𝑒 ℎ 𝑛 + 𝑠𝑒 ℎ 𝑛−2𝑠 .
After simplifying, we get
Corollary 2.15. When 𝑛 > 4𝑠𝑒2Φ,
𝑁𝑆 (𝐹, ℎ) ≪ 𝑠𝑒Φℎ2/𝑛.
38
With fewer restrictions on the values of 𝑛, Mueller and Schmidt in [MS88] show that
Lemma 2.16 (Mueller and Schmidt). Let 3𝑐2 = (𝑛𝑠2)2𝑠/𝑛 and 𝑐 = 𝑠2𝑒(3300𝑠 log 𝑛)/𝑛3 . When
𝑛 ⩾ 4𝑠,
𝑁 2/𝑛𝑆 (𝐹, ℎ) ≪ 𝑐2ℎ + 𝑐 ℎ1/(𝑛−2𝑠)3 .
If in addition 𝑛 ⩾ 𝑠 log3 𝑠, then
𝑁𝑆 (𝐹, ℎ) ≪ ℎ2/𝑛 + 𝑠2ℎ1/(𝑛−2𝑠) .
In the most general 𝑛 ⩾ 4𝑠 case, we may use the fact that 𝑐2 and 𝑐3 are decreasing
functions of 𝑛 to acquire the corollary
Corollary 2.17 (Mueller and Schmidt). When 𝑛 ⩾ 4𝑠,
𝑁 (𝐹, ℎ) ≪ 𝑠2𝑒825 log3 4𝑠𝑆 ℎ2/𝑛.
Again, when ℎ is small relative to the discriminant of 𝐹, Akhtari and Bengoechea
show in [AB20] that
Lemma 2.18 (Akhtari and Bengoechea). If 𝐹 has discriminant Δ𝐹 and
1
|Δ | 8(𝑛−1)
0 𝐹< ℎ < ,
(3𝑛800 log2 𝑛)𝑛/2(𝑛𝑠)2𝑠+𝑛
then
𝑁𝑆 (𝐹, ℎ) ⩽ 12𝑠 + 16.
2.5 Technical Results
The main technical accomplishment of this chapter is the following version of a
counting technique often used in conjunction with “The Gap Principle.”
Lemma 2.19. Suppose that 𝐿, 𝑀,𝑇, 𝑝, 𝑦0, . . . , 𝑦ℓ are positive real numbers satisfying the
following conditions:
1. 𝐿 ⩽ 𝑦0 ⩽ . . . ⩽ 𝑦ℓ ⩽ 𝑀
2. 𝑝 > 2
3. 𝐿𝑝−2 > 𝑇
39
4. 𝑝−1𝑦𝑖+1 ⩾ 𝑇−1𝑦 for each 0 ⩽ 𝑖 < ℓ𝑖
Then [
log(𝑀𝑇−1/(𝑝−2)
]
log )log(𝐿𝑇−1/(𝑝−2))
ℓ ⩽ .
log(𝑝 − 1)
The purpose of this lemma is to bound the number of real numbers 𝑦0, . . . , 𝑦ℓ that
could live between two fixed bounds 𝐿 and 𝑀 under certain assumptions on how far apart
𝑦0, . . . , 𝑦ℓ must be. If, for instance, we knew that 𝐿 ⩽ 𝑦0 < 𝑦1 < · · · < 𝑦ℓ ⩽ 𝑀 and we
knew that there were a 𝛿 > 0 so that 𝑦𝑖 ⩾ 𝑦𝑖−1 + 𝛿 (the condition giving the “gap”), then we
would know that ℓ ⩽ 𝑀−𝐿+1 . Lemma 2.19 instead counts the number of 𝑦𝑖 which could𝛿
live between 𝐿 and 𝑀 under the gap condition that −1 𝑝−1𝑦𝑖+1 ⩾ 𝑇 𝑦 .𝑖
Lemma 2.19 is comparable to Lemma 1 in [SS17]. However, by fixing any 𝐿 > 0,
𝑝 > 2, ℓ ∈ Z>0, and 0 < 𝑇 < 𝐿𝑝−2, then setting 𝑦0 = 𝐿, = −1 𝑝−1𝑦𝑖 𝑇 𝑦 −1 , and 𝑀 = 𝑦ℓ, one𝑖
can see that this upper bound is sharp where the upper bound in Lemma 1 of [SS17] is not.
Proof. We first argue by induction that for each 1 ⩽ 𝑖 ⩽ ℓ, we have
∑ (𝑝−1)
𝑖
𝑦
⩾ ℓ−𝑖𝑦ℓ 𝑖−1 . (2.6)
𝑗=0 (𝑝−1) 𝑗𝑇
This is clearly true for 𝑖 = 0 so now suppose that it is true for generic 0 ⩽ 𝑖 < ℓ. Then
we have
(𝑝−1)𝑖
𝑦
⩾ ∑ ℓ−𝑖𝑦ℓ ( 𝑖−1 (𝑝)−1) 𝑗𝑇 𝑗=0
𝑝−1 (𝑝−1)𝑖
𝑦
ℓ−𝑖−1
⩾ ∑𝑇𝑖−1
𝑗=0 (𝑝−1) 𝑗𝑇
∑(𝑝−1)
𝑖+1
𝑦
= ℓ−𝑖−1
𝑖
𝑗=0 (𝑝−1) 𝑗𝑇
which completes the induction.
Hence, we can take 𝑖 = ℓ in inequality (2.6) to get
𝑀 ⩾ 𝑦ℓ
ℓ
⩾ ∑ (𝑝−1)𝑦0ℓ−1 𝑗
𝑇 𝑗=0
(𝑝−1)
40
(𝑝−1)ℓ
𝑦
= 0
(𝑝−1)ℓ−1
𝑇 𝑝−2
𝐿 (𝑝−1)
ℓ
⩾ .
(𝑝−1)ℓ−1
𝑇 𝑝−2
We then multiply both sides of
𝐿 (𝑝−1)
ℓ
𝑀 ⩾
(𝑝−1)ℓ−1
𝑇 𝑝−2
by 𝑇−1/(𝑝−2) to get ( ) ℓ
𝑀𝑇−1/(𝑝−2) −1/(
(𝑝−1)
⩾ 𝐿𝑇 𝑝−2) .
Taking a log on both sides (and usi(ng the fact th)at 𝐿𝑇−1/(𝑝−2) > 1) yields
log (𝑀𝑇−1/(𝑝−2) ℓ
log 𝐿𝑇−1/( −2)
) ⩾ (𝑝 − 1)
𝑝
and taking logs again and using the fact that 𝑝 > 2 yields the desired inequality. □
There is another technical lemma that we will regularly use in our later estimations. In
the following lemma, the condition that 𝑔(𝑥) > 1 + 𝛿 is necessary and it adds unfortunate
complication to the statements of some later results.
Lemma 2.20. Suppose that 𝑓 (𝑥) and 𝑔(𝑥) are functions R→ R>0 and that there exist
absolute constants 𝛿, 𝑘 > 0 so that 𝑔(𝑥) > 1 + 𝛿 for all 𝑥 and 𝑓 (𝑥) ≪ 𝑔(𝑥)𝑘 . Then
log 𝑓 (𝑥) ≪ log 𝑔(𝑥).
Proof. The statement 𝑓 (𝑥) ≪ 𝑔(𝑥)𝑘 means that there exists 𝑐 so that 𝑓 (𝑥) ⩽ 𝑐𝑔(𝑥)𝑘 .
Taking logs of both sides gives log 𝑓 (𝑥) ⩽ 𝑘 log 𝑔(𝑥) + log 𝑐 and dividing by log 𝑔(𝑥) gives
log 𝑓 (𝑥) 𝑘 log 𝑔(𝑥) + log 𝑐 log 𝑐 log 𝑐
( ) ⩽ ( ) = 𝑘 + ( ) ⩽ 𝑘 +log 𝑔 𝑥 log 𝑔 𝑥 log 𝑔 𝑥 log(1 + 𝛿)
which implies that log 𝑓 (𝑥) ≪ log 𝑔(𝑥). □
We next prove a technical inequality that we will use later.
Lemma 2.21. Suppose that 𝑎, 𝑏 > 0. Then log(𝑎 + 𝑏) ⩽ log+ 𝑎 + log+ 𝑏 + log 2.
41
Proof. Suppose that 𝑎 < 1. If 𝑏 < 1, then log(𝑎 + 𝑏) < log(2) and we are done. Hence,
we may assume that 𝑏 ⩾ 1. Then we have
log(𝑎 + 𝑏) ⩽ log(1 + 𝑏) ⩽ log(2𝑏) = log 𝑏 + log 2 ⩽ log+ 𝑎 + log+ 𝑏 + log 2
and we are done. A similar argument works if 𝑏 < 1, so assume that 𝑎, 𝑏 ⩾ 1 for the rest
of the proof.
Note that
𝑎 + 𝑏 ⩽ 2 max(𝑎, 𝑏) ⩽ 2𝑎𝑏
and taking logs on both sides yields the desired result. □
Finally, we separate out another technical lemma that will be useful exactly once later.
Lemma 2.22. Suppose that 𝐴,𝐶 > 0, 𝐵 ⩾ 1 and that 𝐴𝐶 > 1. Let 𝜀1, 𝜀2 > 0 and suppose
further that 𝐴𝜀2/𝜀1 ⩽ 𝐶. Then
log(𝐴𝐵𝜀1) 𝜀1
⩽ .
log(𝐴𝐵𝜀1+𝜀2𝐶) 𝜀1 + 𝜀2
Proof. Observe that since 𝐴𝜀2/𝜀1 ⩽ 𝐶, we have
𝜀1+𝜀2
𝐴 𝜀1 ⩽ 𝐴𝐶
and hence,
𝜀1+𝜀2
𝐴 𝜀1 𝐵𝜀1+𝜀2 ⩽ 𝐴𝐵𝜀1+𝜀2𝐶.
Taking logs on both sides yields
𝜀1+𝜀2
log(𝐴 𝜀1 𝐵𝜀1+𝜀2) ⩽ log(𝐴𝐵𝜀1+𝜀2𝐶)
and using the facts that 𝐴𝐶 > 1 and 𝐵 ⩾ 1 to conclude that the log on the right-hand side
is positive, we then con(clude)that ( 𝜀1+𝜀2 )
𝜀1+𝜀2 log(𝐴𝐵𝜀1) log 𝐴 𝜀1 𝐵𝜀1+𝜀2
𝜀1
log( + = ⩽ 1.𝐴𝐵𝜀1 𝜀2𝐶) log (𝐴𝐵𝜀1+𝜀2𝐶)
Therefore,
log(𝐴𝐵𝜀1) 𝜀1
log( ⩽𝐴𝐵𝜀1+𝜀2𝐶) 𝜀1 + 𝜀2
as desired. □
42
2.6 Asymptotic Results
2.6.1 Improved Bounds on the Number of Medium Solutions to Thue’s Inequality
In this section, we focus on improving the bounds on the number of primitive medium
solutions to (2.1). This will eventually allow us to prove Theorem 1.16 by using our new
bounds for the number of primitive medium solutions along with others’ bounds for the
number of primitive small and large solutions.
The least restrictive claim we are able to make about the number of primitive medium
solutions to (2.1) is the following:
Lemma 2.23. Let 𝑁𝑀 (𝐹, ℎ) be the number of primitive medium solutions of
|𝐹 (𝑥, 𝑦) | ⩽ ℎ. Suppose further that 𝑛 ⩾ 3𝑠. T(hen )
log 3/2 + log ℎ𝑛 max(1,log𝐻)
𝑁𝑀 (𝐹, ℎ) ≪ 𝑠 · log( 𝑛 − ) .1
𝑠
After showing this lemma, we will show:
Lemma 2.24. In the region of the 𝑛𝑠-plan(e cut out by 𝑛 ⩾ 3𝑠), 𝑠 ⩾ 1, the function
log 3/2 + log ℎ𝑛 max(1,log𝐻)
𝑔(𝑛, 𝑠) = ( )
log 𝑛 − 1
𝑠
has 𝜕𝑔 < 0. Hence, for any fixed 𝑠 and subset 𝐼 ⊆ [3𝑠,∞) with 𝐼 having a minimal
𝜕𝑛
element 𝑛0,
max 𝑔(𝑛, 𝑠) = 𝑔 (𝑛0, 𝑠) .
𝑛∈𝐼
From this, we will acquire a number of corollaries, including the following theorem,
which we will immediately prove.
Theorem 2.25. Let 𝐹 (𝑥, 𝑦) be an irreducible integral binary form of degree 𝑛 ⩾ 3 which
has 𝑠 + 1 nonzero coefficients. Let 𝐻( be a po(sitive integer. Then if 𝑛))⩾ 3𝑠,
𝑁𝑀 (𝐹, ℎ) ≪ 𝑠 1 +
log ℎ
log 𝑠 + ( ( )) . (2.7)max 1, log𝐻 𝐹
Proof. By Lemma 2.24 with 𝐼 = [3𝑠,∞), we can substitute 𝑛 = 3𝑠 into the result from
Lemma 2.23 and follow that with an applicatio(n of Lemma 2.20 us)ing
𝑓 (𝑠) = 33/2 3/2 + log ℎ𝑠 ( ) , 𝑔( ) +
log ℎ 3
𝑠 = 𝑒 𝑠 , 𝑘 = , and 𝛿 = 1.
max 1, log𝐻 max(1, log𝐻) 2
43
This yields
log(33/2𝑠3/2 + log ℎ
( ) ≪ ·( ( max(1,log
)
𝐻)
𝑁𝑀 𝐹, ℎ 𝑠 log 2 ))
≪ + + log ℎ𝑠 1 log 𝑠 ( ) .max 1, log𝐻
□
In the context of trying to prove Conjecture 2.6 and in analogue to Lemma 2.13, we
can say
Corollary 2.26. In addition to the hypotheses of Theorem 2.25, suppose that ℎ ⩽ 𝐻. Then
𝑁𝑀 (𝐹, ℎ) ≪ 𝑠(1 + log 𝑠).
This means that in pursuit of Conjecture 2.6, one only need prove the conjecture for
small solutions.
Corollary 2.27. In addition to the hypotheses of Theorem 2.25, suppose that 𝑛 ⩾ 𝑠1+𝜀
where 𝜀 > 0. Then ( ( ))
𝑁𝑀 (𝐹, ℎ) ≪𝜀 𝑠 1 +
log ℎ
log
max( .1, log𝐻)
Proof. By Lemma 2.24, we can substitute 𝑛 = 𝑠1+𝜀 into the result from Lemma 2.23. □
Corollary 2.28. In addition to the hypotheses of Theorem 2.25, suppose that 𝑛 ⩾ 𝑠1+𝜀 and
ℎ ⩽ 𝐻. Then
𝑁𝑀 (𝐹, ℎ) ≪𝜀 𝑠.
Note that these corollaries together give a strict improvement on Lemma 2.12 when
𝑛 < 𝑠4 and they give a comparable result when 𝑛 ⩾ 𝑠4. Also observe that this corollary
yields the expected heuristic for medium solutions from our discussion before Conjecture
2.5 and from the statement of Conjecture 2.6.
In other contexts (with other assumptions), we can acquire more specific results. In
direct comparison to Lemma 2.11, we have
Corollary 2.29. Under the assumption( that 𝑛 >( 4𝑠𝑒2Φ, ))
𝑁𝑀 (𝐹, ℎ) ≪
𝑠 + + log ℎ1 log 𝑠 .
Φ max(1, log𝐻)
44
Proof. If 𝑛 ⩽ 𝑠2, then from Lemma 2.23, we h(ave )
log 3 + log ℎ𝑠 max(1,log𝐻)
𝑁𝑀 (𝐹, ℎ) ≪ 𝑠 · .log( 𝑛 − 1)
𝑠
Now we can apply the fact that 𝑛 > 4𝑠𝑒Φ to acqu(ire )
1 + log + log ℎ𝑠 max(1,log𝐻)
𝑁𝑀 (𝐹, ℎ) ≪ 𝑠 · .Φ
√
If 𝑛 > 𝑠2, then 𝑛 > 𝑛 and we have
𝑠 ( )
log 𝑛3/2 + log ℎmax(1,log𝐻)
𝑁𝑀 (𝐹, ℎ) ≪ 𝑠 · √ .log( 𝑛 − 1)
Now substituting 𝑛 = m(ax(4𝑠𝑒
2Φ, 𝑠2), usin)g the fact that Φ ≪ max(1, log 𝑠), andobserving that
+ log
3
ℎ log ℎ
𝑠 ( ) ⩾ 𝑠
3 + ,
max 1, log𝐻 ( max(1, log𝐻)we find )
1 + log + log ℎ𝑠 max(1,log𝐻)
𝑁𝑀 (𝐹, ℎ) ≪ 𝑠 · .Φ
□
2.6.2 Proofs of Lemmas 2.23 and 2.24
Before we prove Lemmas 2.23 and 2.24, it is helpful to set up some initial estimates.
√
Lemma 2.30. log𝑌𝐿 ≪ log𝐻 + log ℎ1/𝑛 + 𝑛.
√ √
Proof. We have 𝜆 ≍ 𝑛 and 𝑛 − 𝜆 ≍ 𝑛 − 𝑛, so we can conclude that
1 ( ) + 𝜆log𝑌𝐿 = − log 2𝐶 2 log(4𝐻 (𝑛 + 1)
1/2√︁𝑒𝑛/2)𝑛 𝜆 (𝑛 − 𝜆)𝑎1 1 𝑛
= − log 2 + − log(𝑅ℎ) + − log(2𝐻 𝑛(𝑛 + 1))+𝑛 𝜆 𝑛 𝜆 𝑛 𝜆
+ 𝜆 log(4𝐻 (𝑛 + 1)1/22 √︁𝑒𝑛/2)(𝑛 − 𝜆)𝑎
≪ + log 𝑅 + log ℎ1 √ √ + log𝐻 + log( 𝑛(𝑛 + 1))+
𝑛 − 𝑛 𝑛 − 𝑛
45
√ √
+ 𝑛 + 𝑛
√
( + ) + 𝑛
3/2
√ log𝐻 √ log 𝑛 1 √ .
𝑛 − 𝑛 𝑛 − 𝑛 2(𝑛 − 𝑛)
√
We next apply the fac√︁ts that √𝑛− log𝐻 ≪ log𝐻 and√ √ 𝑛 𝑛
√𝑛
− log( 𝑛 + 1) ≪ log( 𝑛(𝑛 + 1)). Then we have𝑛 𝑛
≪ + log
3 𝑛 + log ℎ
√︁ 𝑛3/2
log𝑌𝐿 1 √ √ + log𝐻 + log( 𝑛(𝑛 + 1)) + √ .
𝑛 − 𝑛 𝑛 − 𝑛 2(𝑛 − 𝑛)
Using the fact that
log3 𝑛 √︁ 𝑛3/2 √
1, √ , log( 𝑛(𝑛 + 1)), √ ≪ 𝑛,
𝑛 − 𝑛 2(𝑛 − 𝑛)
we get
log𝑌 ≪ log𝐻 + log ℎ1/
√
𝑛
𝐿 + 𝑛.
□
Using this result, we conclude that there exists an absolute constant 𝑐 so that
√
𝑌 ⩽ (𝐻ℎ1/𝑛 𝑛𝐿 𝑒 )𝑐 . (2.8)
Now we prove Lemma 2.23
Proof of Lemma 2.23. We want to apply Lemma 2.19 to count primitive medium solutions
to |𝐹 (𝑥, 𝑦) | ⩽ ℎ. Saradha and Sharma’s Lemma 4.4 in [SS17] gives that there exists a set
𝑆 of roots of 𝐹 (𝑥, 1) and a set 𝑆∗ of roots of 𝐹 (1, 𝑦) with cardinalities |𝑆 |, |𝑆∗ | ⩽ 6𝑠 + 4 so
that any solution (𝑥, 𝑦) of (2.1) with min( |𝑥 |, |𝑦 |) ⩾ 12𝑒Ψ (𝑛𝑠)2𝑠/𝑛ℎ1/𝑛 either has𝛼 − 𝑥 𝑦  𝐾⩽ 2|𝑦 |𝑛/𝑠
or  ∗ − 𝑦𝛼  𝐾⩽
𝑥 2|𝑥 |𝑟/𝑠
for some 𝛼 ∈ 𝑆 or 𝛼∗ ∈ 𝑆∗.
Since 𝑌 ⩾ 12𝑒Ψ (𝑛𝑠)2𝑠/𝑛𝑆 ℎ1/𝑛, Saradha and Sharma’s Lemma 4.4 applies to any
medium solution to (2.1). Fix an 𝛼 ∈ 𝑆 and enumerate all of the primitive medium
solutions (𝑥𝑖, 𝑦𝑖) which satisfy  − 𝑥 𝛼 𝑦  𝐾⩽ (2.9)2|𝑦 |𝑛/𝑠
46
so that 𝑌𝑆 ⩽ |𝑦0 | ⩽ |𝑦1 | ⩽ · · · ⩽ |𝑦𝑡 | ⩽ 𝑌𝐿 .For any 0⩽ 𝑖 < 𝑡, we can conclude that𝐾 𝑥𝑖 − 𝑥𝑖+1⩾ |𝑦 |𝑛/𝑠𝑖  𝑦 𝑖 𝑦𝑖+1 𝑥 𝑦
=  𝑖 𝑖+1 − 𝑦 𝑥 +1 𝑖 𝑖𝑦 𝑦 + 𝑖 𝑖 1
1
⩾ | .𝑦𝑖𝑦𝑖+1 |
Rearranging yields “The Gap Principle,”
|𝑦 |𝑛/𝑠−1|𝑦𝑖+1 | 𝑖⩾
𝐾
and we are now in a position to apply Lemma 2.19 with 𝐿 = 𝑌𝑆, 𝑀 = 𝑌𝐿 , 𝑇 = 𝐾 , and 𝑝 = 𝑛 .𝑠
We first observe that 𝑝 > 2 because 𝑛 ⩾ 3𝑠. Next, we have that 𝐿𝑝−2 > 𝑇 because
− − 𝑛−2𝑠𝐿𝑝 2𝑇 1 = 𝑌 𝑠 𝐾−1 (2.10)
𝑆
(12𝑒Ψ)𝑛/𝑠𝑅2ℎ1/𝑠
=
2𝑅(𝑛𝑠)2(12𝑒Ψ)𝑛/ 1𝑠ℎ1/𝑠𝐻 − 1𝑛 𝑠
𝑅 1
= 𝐻 −
1
𝑠 𝑛
2(𝑛𝑠)2
800 log2𝑛 𝑛 1− 1⩾ 𝑠 𝑛4 𝐻 (2.11)2𝑛
> 1.
Hence, we can apply Lemma 2.19 and we find that if there are 𝑡 + 1 primitive medium
solutions which give good rational approxi[mations of 𝛼], then𝑛
log log(
𝑠 −2𝑌 𝐾−1)
𝐿
𝑛
log( −2𝑌 𝑠 𝐾−1)
𝑆
𝑡 ≪ ( 𝑛 − ) . (2.12)log 1
𝑠
To get an upper bound on the right-hand side of this inequality, it is easiest to
manipulate the individual pieces one at a time.
Handling the ca(se where 𝐻) > 1 fi(rs(t, we can a)pply equation)(2.8) to get𝑛 √ ( 𝑛−2)𝑐 1 1
log −2𝑌 𝑠 𝐾−1 ⩽ (lo(g
𝑠
𝐻)ℎ1/𝑛𝑒 𝑛 ) 𝐻 −𝑠 𝑛𝐿 𝑛 1 1
= − 2 𝑐 + − log𝐻+
𝑠 𝑠 𝑛
47
( )
𝑛 − 2 𝑐 ( )
+ 𝑠 𝑛
√
log ℎ + 𝑐 − 2 𝑛. (2.13)
𝑛 𝑠
𝑛
To get a lower bound on log( −2𝑌 𝑠 𝐾−1), we note that equations (2.10) and (2.11) imply
𝑆
that ( )
𝑛 ( )
log( −2 −1) 1 − 1 𝑛 − 𝑠𝑌 𝑠 𝐾 ⩾ log𝐻 = log𝐻. (2.14)
𝑆 𝑠 𝑛 𝑛𝑠
Now we com(bine equa)tions( ((2.13))and (2.14)( 𝑛− ) )
to find
𝑛 ( ) √
log 2𝑌 𝑠 𝐾−1 𝑛 − 2 𝑐 + 1 − 1 log( +) ( −2𝑠 )𝑐𝐻 log ℎ + 𝑐 𝑛 − 2 𝑛𝐿 𝑠 𝑠 𝑛 𝑛 𝑠𝑛 ⩽ 𝑛−𝑠
log −2𝑌 𝑠 𝐾−1 log𝐻𝑛𝑠
𝑆
√
≪ 𝑛(𝑛 − 2𝑠)𝑐 + 𝑛 − 𝑠 + (𝑛 − 2𝑠)𝑐 log ℎ + 𝑐𝑛(𝑛 − 2𝑠) 𝑛
𝑛 − 𝑠 𝑛 − 𝑠 log𝐻 𝑛 − 𝑠
≪ 𝑛3/2 + log ℎ
log𝐻
and inserting this result along with a use of[ Le(mma 2.20] into equation (2.12) yields
log( 𝑛𝑠 −2 )𝑌 𝐾−1log 𝐿𝑛 )
log( 𝑠 −2𝑌 𝐾)−1𝑠𝑡 ≪
l(og 𝑛 − 1𝑠
log 3( )/2 + log ℎ𝑛 log≪ ) 𝐻 .
log 𝑛 − 1
𝑠
Next, we handle the 𝐻 = 1 case. Here, we return to equations (2.10), (2.11), (2.12),
and (2.13) with 𝐻 = 1 and again [use(Lemma 2].20 to find𝑛
log( 𝑠 −2 )𝑌 𝐾−1log 𝐿( 𝑛log 𝑠 −2 ) )𝑌 𝐾−1𝑡 ≪ 𝑆
log (
𝑛 − 1
𝑠
𝑛 )
 ( √ )𝑐( −2) log
1/𝑛 𝑛 𝑠 ℎ 𝑒 
log  800 lo(g3 𝑛−log)(2𝑛4) ≪ [ l(og 𝑛 −) 1𝑠
1 2 ( ) √ ( ) ]log 𝑐 − log(ℎ) + 𝑐 𝑛 𝑛 − 2𝑠 𝑛 𝑠≪
log 𝑛 − 1
𝑠
48
( ( ) )log log(ℎ) + 𝑛3/2≪ .
log 𝑛 − 1
𝑠
For every 𝐻 then, we conclude that ( )
log 𝑛3/2 + log ℎmax(1,log𝐻)
𝑡 ≪ ( ) .
log 𝑛 − 1
𝑠
By Lemma 2.24 (whose proof is indep(endent of this proo)f), the quantity
log log ℎ𝑛3/2 +( max(1),log𝐻)𝑔(𝑛, 𝑠) =
log 𝑛 − 1
𝑠
decreases to 3/2 as 𝑛 →∞ for any fixed 𝑠. In particular, there exists a 𝛿 > 0 so that
𝑔(𝑛, 𝑠) > 𝛿 for all 𝑠 ⩾ 1 and 𝑛 ⩾ 3𝑠, implying that we can safely conclude 𝑡 + 1 ≪ 𝑔(𝑛, 𝑠).
The upper bound 𝑡 + 1 ≪ 𝑔(𝑛, 𝑠) on the number of medium solutions which yield
good rational approximations to 𝛼 in the sense of (2.9) is independent of 𝛼. The same
bound on the number of medium solutions which yield good rational approximations of
any 𝛼∗ ∈ 𝑆∗ holds by symmetry. Since |𝑆 | + |𝑆∗ | ≪ 𝑠 and since we have an upper bound
on the number of good rational approximation(s to 𝛼 ∈ 𝑆 and 𝛼∗)∈ 𝑆∗, we conclude that
log 3/2 + log ℎ𝑛 max(1,log𝐻)
𝑁𝑀 (𝐹, ℎ) ≪ 𝑠 · log( 𝑛 − 1) .
𝑠
□
Next, we proceed to prove Lemma 2.24.
Proof of Lemma 2.24. For ease of notation, set = log ℎ𝐿 max(1,log𝐻) so we have
( ) log(𝑛
3/2 + 𝐿)
𝑔 𝑛, 𝑠 = .
log( 𝑛 − 1)
𝑠
Note that we have
𝜕𝑔 log(
𝑛 − 1) 3𝑛1/22( 3/2+ ) − log(𝑛
3/2 + 𝐿) 1 1
𝑠 𝑛𝑛 𝐿 −1 𝑠
= 𝑠
𝜕𝑛 log2( 𝑛 − 1)
𝑠
3 log( 𝑛 − 1) (𝑛 − 𝑠) − 2(𝑛 + 𝐿𝑛−1/2) log(𝑛3/2 + 𝐿)
= 𝑠
2(𝑛 + 𝐿𝑛−1/2) (𝑛 − 𝑠) log2( 𝑛 − 1)
𝑠
log(( 𝑛 − 1)3) (𝑛 − 𝑠) − (𝑛 + 𝐿𝑛−1/2) log((𝑛3/2 + 𝐿)2)
= 𝑠 .
2(𝑛 + 𝐿𝑛−1/2) (𝑛 − 𝑠) log2( 𝑛 − 1)
𝑠
49
Since the denominator in the above expression is positive, 𝜕𝑔 has the same sign as its
𝜕𝑛
numerator. Upon checking that ( 𝑛 − 1)3 < (𝑛3/2 + 𝐿)2 and observing that
𝑠
𝑛 − 𝑠 < 𝑛 + 𝐿𝑛−1/2, we find that 𝜕𝑔 < 0. □
𝜕𝑛
2.6.3 Improving Akhtari and Bengoechea’s Medium Solution Bound
In the context of trying to prove something like Conjecture 2.6, Akhtari and
Bengoechea have changed the conditions slightly so that ℎ is assumed to be small relative
to |Δ𝐹 | rather than being small relative to 𝐻 (𝐹). These new assumptions on ℎ yield
additional information about the size of 𝑌𝐿 , which they leverage to improve bounds on the
number of primitive solutions. Our approach to bounding the number of medium solutions
can be applied to their context and we have the following slight improvement of
Proposition 3.2 in [AB20].∗ The proof of this theorem will also use a different technique
than Akhtari and Bengoechea used in their paper.
Theorem 2.31. Suppose that ℎ satisfies
1
|𝐷 | 8(𝑛−1)
0 < ℎ <
(2𝑛800 log2 𝑛)𝑛/2(𝑛𝑠)2𝑠+𝑛
and that 𝑛 > 3𝑠. Then
𝑁𝑀 (𝐹, ℎ) ≪ 𝑠 log(𝑠).
In addition, if 𝑛 ⩾ 𝑠1+𝜀 for some 𝜀 > 0, then
𝑁𝑀 (𝐹, ℎ) ≪𝜀 𝑠.
Proof. The first several lines of this proof are identical to the first several lines of the proof
of Lemma 2.23. We modify the proof beginning at the observation that if there are 𝑡 + 1
primitive medium solutions corresponding[ to a root 𝛼 o]f 𝐹 (𝑥, 1) or 𝐹 (1, 𝑦), then𝑛
log log(
𝑠 −2𝑌 𝐾−1)
𝐿
𝑛
log( 𝑠 −2𝑌 𝐾−1)
𝑆
𝑡 ≪ .
log( 𝑛 − 1)
𝑠
∗The improvement lies in reducing the assumption that 𝑛 > 10𝑠 to 𝑛 > 3𝑠 and also in
the fact that the definition of a medium solution in this paper is slightly broader than the
definition of a medium solution in [AB20].
50
We now apply equation (17) from [AB20], namely that 𝑌 2𝐿 ≪ 𝐻 , so that there exists a
constant 𝑐 so that 𝑌𝐿 ⩽ 𝑐𝐻2: [ 𝑛 −2 ]log(𝑌 𝑠 𝐾−1log )𝐿𝑛
log( 𝑠 −2𝑌 𝐾−1[ )𝑆𝑡 ≪ log( 𝑛 − 1)𝑠 ]
( 2) 𝑛log 𝑐𝐻 𝑠 −2log 𝐾
−1
𝑛
log( −2𝑌 𝑠 𝐾)−1⩽ 𝑆 . (2.15)
log 𝑛 − 1
𝑠
𝑛
In the case that 𝐻 = 1, we can use equation (2.11) to obtain that −2 796𝑌 𝑠 𝐾−1 ⩾ 𝑛 so
that [ ] [ ] 𝑆
2
𝑛
log log 𝑐 𝑠
−2𝐾−1 ( 𝑛log −2) log(𝑐)𝑠𝑛
log 𝑠 −2𝑌 𝐾−1 log( 𝑛7962 ) log(( 𝑛 − 2) log(𝑐))≪ 𝑆 ⩽ ⩽ 𝑠𝑡 ≪ 1.
log( 𝑛 − 1) log( 𝑛 − 1) log( 𝑛 − 1)
𝑠 𝑠 𝑠
In the case that 𝐻 > 1, we return[ to (equation (2.15) )] to get
log (𝑐𝐻2) 𝑛 −2 1𝑠 𝐻 𝑠 − 1𝑛
log ( 1 1log𝐻 𝑠 − 𝑛𝑡 ≪ [ )log 𝑛 − 1𝑠 ]
log (
𝑛−2) log(𝑐) 2𝑛−4+ 1− 1
𝑠 𝑠 𝑠 𝑛
( 1− 1 ) log( ) +𝐻 1− 1
[ 𝑠 𝑛 𝑠 𝑛= log( 𝑛 − 1)𝑠 ]
( 2𝑛log −4) log(𝑐)
2𝑛−4
𝑠 𝑠
1− 1 + 1− 1 + 1
[ 𝑠 𝑛 𝑠 𝑛⩽ log( 𝑛 − 1)𝑠 ]
log (log(𝑐) + 1) 2𝑛(2𝑛−2𝑠) + 1
𝑛−𝑠
⩽
log( 𝑛 − 1)
𝑠
≪ log(𝑛)
log( 𝑛 − 1) .
𝑠
Defining 𝑔(𝑛, 𝑠) := log(𝑛)log( 𝑛−1) , we can show as in the proof of Lemma 2.23 that
𝑠
𝑡 + 1 ≪ 𝑔(𝑛, 𝑠) also holds. Additionally, for fixed 𝑠, the function 𝑔(𝑛, 𝑠) is minimized
when 𝑛 = 3𝑠. Since there are |𝑆 | + |𝑆∗ | ≪ 𝑠 roots 𝛼 of 𝐹 (𝑥, 1) or 𝐹 (1, 𝑦) to which
primitive medium solutions produce good rational approximations, we may conclude that
𝑁𝑀 (𝐹, ℎ) ≪ 𝑠 · 𝑔(𝑛, 𝑠) ⩽ 𝑠 · 𝑔(3𝑠, 𝑠) ·
log(3𝑠)
= 𝑠
log( ) ≪ 𝑠 log(𝑠).2
51
Likewise, an identical argument will show that 𝑔(𝑛, 𝑠) is minimized at 𝑛 = 𝑠1+𝜀 when
𝑛 ⩾ 𝑠1+𝜀. In the case when 𝑛 ⩾ 𝑠1+𝜀 then, we acquire
1+𝜀
𝑁𝑀 (𝐹, ℎ) ≪ 𝑠 · ( 1+𝜀 ) ·
log(𝑠 )
𝑔 𝑠 , 𝑠 = 𝑠 ≪ 𝑠.
log( 𝜀 − 1) 𝜀𝑠
□
2.6.4 Proof of Theorem 1.16
Finally, we want to prove our main asymptotic theorem, Theorem 1.16.
Proof. Suppose that 𝑛 > 4𝑠𝑒2Φ. From Lemma 2.10, we have that
𝑁𝐿 (𝐹, ℎ) ≪ 𝑠.
From Lemma 2.23 and Lemma 2.24, we h(ave that )
log 8 3/2 (3Φ + log ℎ𝑠 𝑒 max()1,log𝐻)𝑁𝑀 (𝐹, ℎ) ≪ 𝑠 .log 4𝑒2Φ − 1
Finally, from Lemma 2.15, we have
𝑁 (𝐹, ℎ) ≪ 𝑠𝑒Φℎ2/𝑛𝑆 .
Combining these three inequalities y(ields that© )log 8𝑠3/2 3Φ + log ℎ𝑒 max(1,log𝐻) ª
𝑁 (𝐹, ℎ) ≪ 𝑠𝑒Φℎ2/𝑛 «­­ −Φ −2/𝑛 ®(1 + 𝑒 ℎ ( log(4𝑒2Φ − 1) )¬®)
≪ 𝑠𝑒Φℎ2/𝑛 (1 + 𝑒−Φℎ−2/𝑛 log ℎlog 8( 𝑠9/2 + max(1, log𝐻( ) ) ))
≪ 𝑠𝑒Φℎ2/𝑛 1 + log−3(𝑠)ℎ−2/𝑛 log+( 9/2) + + log ℎ8𝑠 log + log 2
max(1, log𝐻)
where the last step uses Lemma 2.21. From this last step, we now see
𝑁 (𝐹, ℎ) ≪ 𝑠𝑒Φℎ2/𝑛
and this concludes the proof. □
52
2.7 Explicit Results for Trinomials
Now consider the particular case where 𝑠 = 2, i.e. 𝐹 (𝑥, 𝑦) is a trinomial. In addition,
we only examine the Thue equation (2.4).
In this section, we follow Thomas in [Tho00] for much of our reasoning. However, we
use different notation: the parameters which Thomas calls 𝑢 and 𝑣, we call 𝑎 and 𝑏† and
the parameters which Thomas calls 𝑏 and 𝑏0, we will call 𝑑 and 𝑑0.
Throughout the remainder of this section, suppose that
𝐹 (𝑥, 𝑦) = ℎ𝑛𝑥𝑛 + ℎ𝑘𝑥𝑘 𝑦𝑛−𝑘 + ℎ 𝑦𝑛0 where ℎ𝑛, ℎ𝑘 , ℎ0, 𝑛, 𝑘 ∈ Z, 0 < 𝑘 < 𝑛, and 𝑛 ⩾ 6.
Suppose further that 𝐹 (𝑥, 𝑦) is irreducible over Z[𝑥, 𝑦]. Let 𝐻 = max( |ℎ𝑛 |, |ℎ𝑘 |, |ℎ0 |) be
the height of 𝐹 (𝑥, 𝑦). Any time we refer to a “solution,” we specifically mean a solution to
equation (2.4) in Z2.
We will not give a sophisticated bound on the number of solutions (𝑝, 𝑞) with
|𝑝𝑞 | ⩽ 1 and we will consider (𝑝, 𝑞) and (−𝑝,−𝑞) to be equivalent solutions, spurring the
following definition.
Definition 2.32. A pair (𝑝, 𝑞) ∈ Z2 is called regular if 𝑝 ≠ 0, 𝑞 > 0, and |𝑝 | ≠ 𝑞.
If there are 𝑟 regular solutions to (2.4), then there will be at most 2𝑟 + 8 distinct
solutions since for every solution (𝑝, 𝑞) with |𝑝𝑞 | > 1, either (𝑝, 𝑞) or (−𝑝,−𝑞) is regular
and there are at most 8 solutions with |𝑝𝑞 | ⩽ 1. From this fact and Theorem 2.33 below,
Theorem 1.17 will follow.
Theorem 2.33. Equation (2.4) has at most 𝑣(𝑛)𝑧(𝑛) regular solutions where 𝑣(𝑛) and
𝑧(𝑛) are defined in Theorem 1.17.
More specifically, let 𝑓 (𝑥) := 𝐹 (𝑥, 1) and set 𝑅𝐹 to be the number of real roots of 𝑓 .
We also wish to include certain critical points, so we make the following definition:
Definition 2.34. A critical point 𝜏 ∈ R of 𝑔(𝑥) ∈ R[𝑥] is proper if there exists a
neighborhood𝑈 of 𝜏 for which 𝑔′′(𝑥)𝑔(𝑥) > 0 for all 𝑥 ∈ 𝑈 \ {𝜏}.
Now let 𝐶𝐹 to be the number of proper critical points of 𝑓 (𝑥). Setting 𝑁𝐹 to be the
number of regular solutions to (2.4), we will show the following theorem.
†This aligns with similar notation used in [AB20] for instance, and also makes clear the
difference between these parameters—whose choice will depend on 𝑛—and the values 𝑢𝑛
to be defined later and 𝑣(𝑛) defined in Theorem 1.17
53
Theorem 2.35. Let 𝐹 (𝑥, 𝑦) be a trinomial of degree 𝑛 ⩾ 6. Then
𝑁𝐹 ⩽ 𝑧(𝑛)𝑅𝐹 + ℓ(𝑛)𝐶𝐹
where ℓ(𝑛) is defined by the following table.
𝑛 6–7 8 ⩾ 9
ℓ(𝑛) 4 3 2
We first show that Theorem 2.35 implies Theorem 2.33. Since ℓ(𝑛) is less than 𝑧(𝑛),
we have that 𝑧(𝑛)𝑅𝐹 + ℓ(𝑛)𝐶𝐹 ⩽ 𝑧(𝑛) (𝑅𝐹 + 𝐶𝐹). Moreover, one can check with calculus
that polynomials with at most four real roots have 𝑅𝐹 + 𝐶𝐹 ⩽ 𝑣(𝑛). Since irreducible
trinomials have at most four real roots, we get Theorem 2.33 from Theorem 2.35.
To prove Theorem 2.35, we need some additional setup.
Definition 2.36. For a polynomial 𝑔(𝑥) ∈ R[𝑥], an exceptional point of 𝑓 is either a real
root or a proper critical point of 𝑔(𝑥)
Let E( 𝑓 ) be the set of exceptional points, 𝜏1 < 𝜏2 < · · · < 𝜏𝑐, of 𝑓 . Note that there
exist improper critical points 𝜂1 < 𝜂2 < · · · < 𝜂𝑐−1 so that
𝜏1 < 𝜂1 < 𝜏2 < 𝜂2 < · · · < 𝜂𝑐−1 < 𝜏𝑐. Setting 𝜂0 = −∞ and 𝜂𝑐 = +∞, we can define
𝐽1 = (−∞, 𝜂1) and 𝐽𝑖 = [𝜂𝑖, 𝜂𝑖+1) for 1 ⩽ 𝑖 ⩽ 𝑐.
Definition 2.37. A real number 𝜌 belongs to 𝜏𝑖 (and 𝜏𝑖 belongs to 𝜌) if 𝜌 ∈ 𝐽𝑖.
Thomas, in [Tho00], shows that the number of regular solutions (𝑝, 𝑞) of (2.4) for
which there exists a critical point of 𝑓 (𝑥) = 𝐹 (𝑥, 1), 𝜏, so that 𝑝 belongs to 𝜏 is no larger
𝑞
than ℓ(𝑛) (see the completion of the proof of Thomas’ Theorem 2.2, given after the
statement of Theorem 7.1). So it only remains to show
Lemma 2.38. The number of regular solutions, (𝑝, 𝑞), of (2.4) for which 𝑝 belongs to a
𝑞
real root of 𝑓 is no larger than 𝑧(𝑛).
By Theorem 2.2 in [Tho00], it suffices to show Lemma 2.38 for real roots of 𝑓 which
are greater than 1. Then by Lemma 2.4 of [Tho00], we conclude that any regular (𝑝, 𝑞) for
which 𝑝 belongs to an exceptional point greater than 1 has 𝑝 > 𝑞 ⩾ 1 and so we may
𝑞
assume that 𝑝 > 𝑞 ⩾ 1. Defining
3 if 6 ⩽ 𝑛 ⩽ 8𝑝0(𝑛) :=  ,2 if 𝑛 ⩾ 9
54
we note that any regular solution (𝑝, 𝑞) with 𝑝 > 𝑞 ⩾ 1 must satisfy
𝑝 ⩾ 𝑝0(𝑛) (2.16)
except for possibly (2, 1) when 𝑛 ⩽ 8.
Definition 2.39. A solution, (𝑝, 𝑞) to equation (2.4) with 𝑝 > 𝑞 ⩾ 1 and 𝑝 ⩾ 𝑝0(𝑛) is
called special.
Since at most one solution is not special in the case that 6 ⩽ 𝑛 ⩽ 8, it suffices to show
the following lemma, which will be our final reduction.
Lemma 2.40. Let 𝛼 > 1 be a real root of 𝑓 (𝑥). Then the number of special solutions
(𝑝, 𝑞) of (2.4) for which 𝑝 belongs to 𝛼 is no greater than 𝑧(𝑛) − 1 if 6 ⩽ 𝑛 ⩽ 8 and no
𝑞
greater than 𝑧(𝑛) if 𝑛 ⩾ 9.
To prove Lemma 2.40, we split solutions into two cases: small and large. For 𝐹 (𝑥, 𝑦)
of degree 𝑛 and height 𝐻 = 𝐻 (𝐹), we choose a constant 𝑌 = 𝐻𝜒𝑛 · 𝑒𝜋𝑛𝐹 (for some values
𝜒𝑛 and 𝜋𝑛 to be specified later, but which depend only on 𝑛) and make the following
definition.
Definition 2.41. A special solution (𝑝, 𝑞) to (2.4) is small if 𝑞 ⩽ 𝑌𝐹 and is large otherwise.
2.7.1 Small Special Solutions
One of Thomas’ main achievements in [Tho00] is the following theorem (numbered
4.1 in [Tho00]), which is a version of the “Gap Principle:”
Theorem 2.42 (Thomas). Suppose that 𝐹 (𝑥, 𝑦) ∈ Z[𝑥, 𝑦] is an irreducible (over Z)
trinomial binary form of degree 𝑛 ⩾ 5 and height 𝐻 = 𝐻 (𝐹). Let (𝑝, 𝑞) and (𝑝′, 𝑞′) be
special solutions to (2.4) which belong to a real root and suppose 𝑞′ > 𝑞. Then
𝑑/𝑛 𝑛∗𝐻 𝑝 −𝑑𝑞𝑑
𝑞′ > (2.17)
𝐾𝑑 (𝑛)
where
∗ 𝑛 − 2𝑛 := , (2.18)
2
𝑑 is chosen to be any real number satisfying 0 ⩽ 𝑑 ⩽ 𝑛∗, and
𝐾𝑑 (𝑛) := 𝑚𝑛 (𝑟 (1 + 𝑢 ))𝑑𝑛 𝑛 (2.19)
55
where √︄ √︄
2𝑛
𝑚 = 2 , 𝑟 = (2.032)1/𝑛 2𝑛 ( − 1) ( − 2) 𝑛 , 𝑢𝑛 =𝑛 𝑛 (𝑛 − 2) . (2.20)𝑝𝑛0
This approximation result will be helpful in proving the following proposition:
Proposition 2.43. Let 𝛼(> 1 be a re)al roo(t of 𝑓 (𝑥). There are no more than ­©
)
­ 𝜋 𝑑 ­­ log
𝜒𝑛𝑛(𝑑−1)+𝑑 log 𝑛 +
𝑑 (𝑑−1)+ 1𝑑 log ( )− −1 𝑑0 (𝑑−1)+𝑑0 𝐾 𝑛 𝑑 𝑄 ®ª
𝑇 :=  𝑑 1max , ®log log ®®
 + 2 (2.21) « 𝑑 𝑑 ¬
small special solutions (𝑝, 𝑞) where 𝑝/𝑞 belongs to 𝛼.
Proof. If there less than 2 special solutions (𝑝, 𝑞) where 𝑝/𝑞 belongs to 𝛼, then we are
done. Otherwise, suppose that there are exactly 𝑡 + 2 small special solutions (𝑝, 𝑞) where
𝑝/𝑞 belongs to 𝛼 and 𝑡 ⩾ 0. Label those 𝑡 + 2 solutions as (𝑝0, 𝑞0), . . . , (𝑝𝑡+1, 𝑞𝑡+1)
ordered so that
1 ⩽ 𝑞0 < 𝑞1 < . . . < 𝑞𝑡+1 ⩽ 𝑌𝐹
(the strict inequality follows from the fact that the 𝑝𝑖 are principal convergents to 𝛼 by
𝑞𝑖
Corollary 3.2 in [Tho00]).
Choose numbers 𝑑0, 𝑑 ∈ R ∗>0 and recall the definition of 𝑛 from equation (2.18)
before making the following definitions:
𝑝 𝑐
:= ∗ − := ( ) := 0(𝑛)
0
𝑐0 𝑛 𝑑0, 𝐾0 𝐾𝑑0 𝑛 , 𝑄1 .𝐾0
In particular, choose 𝑑 and 𝑑0 so that
1 < 𝑑 ⩽ 𝑛∗, (2.22)
0 ⩽ 𝑑0 ⩽ min(𝑛∗ − 1.4, 𝑑), (2.23)
𝑄𝑑−11 > max(1, 𝐾𝑑 (𝑛)). (2.24)
In the proof of Proposition 2.47 and in the computations in Section 2.7.4, we show by
example that choosing such 𝑑 and 𝑑0 are possible.
First, observe that by Theorem 2.42 applied to 𝑞1 > 𝑞0 ⩾ 1 (and using the observation
that 𝑝0 ⩾ 𝑝0(𝑛)), we get 𝑞 𝑑1 > 𝐻 0/𝑛𝑄1.
56
Here is where we depart from Thomas’ method. We now aim to apply Lemma 2.19 to
the inequalities
𝐻𝑑0/𝑛𝑄1 < 𝑞1 < 𝑞2 < · · · < 𝑞𝑡+1 ⩽ 𝑌𝐹
and
𝐻𝑑/𝑛𝑞𝑑
𝑖
𝑞𝑖+1 >
𝐾𝑑 (𝑛)
.
In the notation of Lemma 2.19, we have 𝐿 = 𝐻𝑑0/𝑛𝑄1, 𝑀 = 𝐾𝑑 (𝑛)𝑌𝐹 , 𝑝 = 𝑑 + 1, and 𝑇 = 𝐻𝑑/𝑛 .
To apply the conclusion of Lemma 2.19, we need to check that 𝑝 > 2 (trivial based on the
fact that 𝑑 is chosen to be greater than 1 from (2.22)) and we need to check that 𝐿𝑝−2 > 𝑇 .
But this occurs if and only if ( )𝑑−1 𝐾𝑑 (𝑛)
𝐻𝑑0/𝑛𝑄1 > ,
𝐻𝑑/𝑛
i.e.
𝐻 (𝑑0 (𝑑−1)+𝑑)/𝑛𝑄𝑑−11 > 𝐾𝑑 (𝑛),
which is guaranteed by (2.24).
Now applyingLemm( a 2.19 and u)sing the fact that 𝑡 is an integer yields
⌊ 
( )
𝐾 (𝑛) −
1
𝑑−1
 l(og 𝑌 𝑑𝐹 𝑑/𝑛log 
𝐻( ) )− 1
log 𝑑 /𝑛 𝐾𝑑 (𝑛) 𝑑−1 
 ⌋
𝐻 0 𝑄1
𝐻𝑑/𝑛 
𝑡 ⩽  ( log 𝑑
⌊ 
)
1 𝑑 
 lo(g 𝑌 𝐾 (𝑛)− 𝑑−1 𝐻 𝑛(𝑑−1) log  𝐹 𝑑 )

𝑑 
log ( )−
1 0 + 𝑑
𝐾 𝑛 𝑑−1 𝐻 𝑛 𝑛(𝑑−1)𝑑 𝑄1  ⌋
=
 log 𝑑 ( )− 1 𝑑 log 𝐾 (𝑛) 𝑑−1 𝐻 𝑛(𝑑−1)⌊ log  ( log(𝑌𝐹 ) ) (
𝑑
+ ) 
1 𝑑0 𝑑 1 𝑑

0 𝑑
log 𝐾 (𝑛)− +𝑑−1 𝐻 𝑛 𝑛(𝑑−1)𝑑 𝑄1 log 𝐾𝑑 (𝑛)−
+
𝑑− 1 𝐻 𝑛 𝑛(𝑑−1) 𝑄1  ⌋
= . (2.25)
log 𝑑
We now claim that we can apply (Lemma 2.22 to the)logarithmic quantity
lo( 𝑑g − 1𝐾 𝑛(𝑑−1)𝑑 (𝑛) 𝑑−1𝐻 )
𝑑0 𝑑
log 𝐾 − 1𝑑 (𝑛) 𝑑−1 +𝐻 𝑛 𝑛(𝑑−1)𝑄1
57
with 1𝐴 = 𝐾 − 𝑑 𝑑𝑑 (𝑛) 𝑑−1 , 𝐵 = 𝐻, 𝐶 = 𝑄 01, 𝜀1 = ( −1) , and 𝜀2 = . All of the hypotheses are𝑛 𝑑 𝑛
clear except possibly that 𝐴𝐶 > 1 which follows from (2.24) and 𝐴𝜀2/𝜀1 < 𝐶.
To verify this second hypothesis, we want to first note that
𝜀2 𝑑0(𝑑 − 1)=
𝜀1 𝑑
and so, recalling the definition of 𝐾𝑑 (𝑛) from (2.19),
𝑑
𝐴𝜀2/𝜀1 = 𝐾𝑑 (𝑛)−
0
𝑑 .
𝑑
( 0= 𝑚𝑛 (𝑟𝑛 (1 + 𝑢 𝑑 −𝑛)) ) 𝑑
− 𝑑0
= 𝑚 𝑑𝑛 (𝑟𝑛 (1 + 𝑢 −𝑑0𝑛))
whereas
𝐶 = 𝑄1
𝑝0(𝑛)𝑐0=
𝐾𝑑0 (𝑛)
= 𝑝 (𝑛)𝑛∗−𝑑0𝑚−1(𝑟 (1 + 𝑢 ))−𝑑0 0𝑛 𝑛 𝑛 .
Hence, 𝐴𝜀2/𝜀1 < 𝐶 if and only if
1− 𝑑0
𝑚 𝑑𝑛 < 𝑝0(𝑛)𝑛
∗−𝑑0 .
But this follows from the selection of 𝑑0 ⩽ 𝑑 (by (2.23)) along with the fact that 𝑚𝑛 is a
decreasing function of 𝑛 for 𝑛 ⩾ 6 (which we√︁can see from (2.20)), so1− 𝑑0 ∗
𝑚 𝑑𝑛 < max(1, 𝑚 ) ⩽ 2 3/5 < 21.4 < 𝑝 (𝑛)𝑛 −𝑑0 0𝑛 .
Now that we have verified that we may apply Lemma 2.22, we continue the estimation
we left off in inequality (2.25):
⌊ log 
( )
1 𝑑 
 (
log 𝐾 ( −𝑛) 𝑑−1 𝐻 𝑛(𝑑−1) 
log(𝑌𝐹 )
𝑑 
 ) + ( ) 1 𝑑0log ( )− 𝑛 + 𝑑 𝑑− 1 0 𝑑𝐾 𝑛 𝑑−1 𝐻 𝑛(𝑑−1) 𝑄 log 𝐾 (𝑛) 𝑑−1 𝐻 𝑛 +1 𝑛(𝑑−1) 𝑑 𝑑 𝑄1  ⌋
𝑡 ⩽  log 𝑑
⌊ log  (

 log( ) )
𝑑
𝑌𝐹 + 𝑛(𝑑−1)  − 1 𝑑0 𝑑 𝑑+ 0 + 𝑑log ( ) − 𝑛 ( −1) 𝐾 𝑛 𝑑 1 𝐻 𝑛 𝑑 𝑄1 𝑛 𝑛(𝑑−1)𝑑  ⌋
⩽
log 𝑑
58
⌊ log 


 ( log(𝑌𝐹 ) ) +
𝑑 
− 1
𝑑0 + 𝑑 𝑑0 (𝑑−1)+𝑑log 𝐾 (𝑛) 𝑑−1 𝐻 𝑛 𝑛(𝑑−1)𝑑 𝑄1  ⌋
= .
log 𝑑
Now using the definition 𝑌 = 𝐻𝜒 𝑛 · 𝑒
𝜋𝑛
𝐹 , we have
⌊  log  𝜒𝑛 log𝐻(+𝜋𝑛 ) + 𝑑 𝑑0 (𝑑−1)+𝑑 − 1 𝑑0 (𝑑−1)+𝑑  ⌋𝑛( −1) log𝐻+log 𝐾𝑑 (𝑛) 𝑑−1 𝑄𝑑 1 
𝑡 ⩽
 log 𝑑 ©­
( ) ( ) 
­­ log
𝜒𝑛 𝑑 𝜋𝑛 𝑑 
­ 𝑑0 (𝑑−1)+𝑑
+
𝑑0 (𝑑−1)+ ª𝑑 log 1 + 
max 𝑛(𝑑−1) log ( )
− −1 𝑑0 (𝑑−1)+𝑑𝐾 𝑛 𝑑 𝑄1 ®𝑑 
⩽  , ®log 𝑑 log 𝑑 ®®
 « ©­­ ( )
( )  ¬
 ­ log 𝜒𝑛𝑛(𝑑−1)+𝑑 log
𝜋𝑛 + 𝑑 ª

 ­ 𝑑0 (
1 
𝑑−1)+𝑑
max log ( )
− −1 𝑑0 (𝑑−1)+𝑑𝐾
= 𝑑
𝑛 𝑑 𝑄1 ®

, ®
« log 𝑑 log ®®

𝑑 ¬
= 𝑇 − 2.
Therefore, the number of small special solutions (𝑝, 𝑞) for which 𝑝/𝑞 belongs to 𝛼 is
𝑡 + 2 ⩽ 𝑇 . □
2.7.2 Large Special Solutions
Here we follow Thomas in [Tho00] as he follows Bombieri-Schmidt in [BS87]. If we
choose numbers 𝑎 and 𝑏 satisfying √︂
𝑛 + 𝑎2
0 < 𝑎 < 𝑏 < 1 − 2 ·
𝑛2
(2.26)
then we can √︁define
2(𝑛 + 𝑎2) 𝐿 1 1
𝐿 = 𝐷 = 𝐴 =
1 − 𝑏 𝑛 − 𝐿 𝑎2
𝐸 = 2 2 .2(𝑏 − 𝑎 )
Now we choose
𝜒𝑛 = 𝐷 (𝐴 + 1) + 1, (2.27)
59
( ( + ) + ) ( ) + (𝐷 + 1) log(𝑛) 𝑛𝐴𝐷𝜋𝑛 = 𝐷 4 𝐴 2 log 2 + . (2.28)2 2
With these choices of 𝜋𝑛 and 𝜒𝑛, we aim to apply Lemma 2 of [BS87] and conclude
the following:
Proposition 2.44. Suppose 𝛼 > 1 is a real root of 𝑓 (𝑥). If 𝜒𝑛 ⩾ 2 and
𝜋𝑛 ⩾ 5 log(2) + 2 log(𝑛), then⌊there are at most ⌋
log 𝐸 + 2 log(𝑛) − log(𝐿 − 2)
𝑍 := ( + 2 (2.29)log 𝑛 − 1)
large special solutions belonging to 𝛼.
The proof of this proposition relies on the two following lemmas:
Lemma 2.45. 𝑌𝐹 as defined here is greater than or equal to 𝑌0 as defined in [BS87].
Lemma 2.45 ensures that any large solution in the sense of this paper is a large
solution in the sense of Bombieri and Schmidt.
Lemma 2.46. Suppose that 𝜒𝑛 ⩾ 2 and 𝜋𝑛 ⩾ 5 log(2) + 2 log(𝑛). If 𝛼 > 1 is a real root of
𝑓 (𝑥) and (𝑝, 𝑞) is a large special solution of (2.4) so that 𝑝/𝑞 belongs to 𝛼, then 𝛼 is the
closest (complex) root of 𝑓 (𝑥) to 𝑝/𝑞.
Given an algebraic 𝛽, Lemma 2 of [BS87] only counts the number of rational numbers
which are nearest to 𝛽 out of all of the conjugates of 𝛽 and which form good
approximations of 𝛽. If there were a real root 𝛼 > 1 of 𝑓 (𝑥) and a large special solution
(𝑝, 𝑞) of (2.4) for which 𝑝/𝑞 belonged to 𝛼 yet there was a root 𝛽 of 𝑓 (𝑥) with 𝛽 closer to
𝑝/𝑞 than 𝛼, Lemma 2 of [BS87] would not count 𝑝/𝑞. However, Lemma 2.46 confirms
that this is not the case if we choose 𝑎 and 𝑏 carefully enough to make 𝜒𝑛 ⩾ 2 and
𝜋𝑛 ⩾ 5 log(2) + 2 log(𝑛).
We first prove these two lemmas:
Proof of Lemma 2.45. 𝑌0 depends on the Mahler measure 𝑀 (𝐹) rather than the height
𝐻 (𝐹). These are related (for trinomials 𝐹 (𝑥, 𝑦)) by 𝑀 (𝐹) ⩽ 31/2𝐻 (𝐹), which follows
from the fact that 𝑀 (𝐹) ⩽ ℓ2(𝐹) (see Lemma 1.6.7 in [BG06]). Now, using Thomas’
notation in Bombieri and Schmidt’s notation, we( have) that𝜆
1
: 𝑛−𝜆𝑌0 = (2𝐶) 𝑛−𝜆 4𝑒𝐴1
60
where
𝐶 =√︂(2𝑛1/2𝑀 (𝐹))𝑛,
2
𝑡 = ,
𝑛 + 𝑎2
𝑡2 ( )
𝐴1 = log 𝑀 (𝐹) +
𝑛
,
2 − 𝑛𝑡2 2
2
𝜆 =
𝑡 ( .1 − 𝑏)
Some of our other constants regularly appear in the estimation which shows 𝑌0 < 𝑌𝐹
and we list them here for simplicity:
1 2 2 1 𝑡2
𝐴 = √︁2 = = · = ,𝑎 2(𝑛 + 𝑎2) − 2𝑛√︂ 𝑛 + 𝑎2 2 − 2𝑛 2𝑛+𝑎2 2 − 𝑛𝑡
2(𝑛 + 𝑎2) 2 𝑛 + 𝑎2 2
𝐿 = − = − = = 𝜆,1 𝑏 1 𝑏 2 𝑡 (1 − 𝑏)
𝐿 𝜆
𝐷 = − = .𝑛 𝐿 𝑛 − 𝜆
Note also that this implies that 𝐷 + 1 = 𝑛
𝑛− .𝜆
Before making the final estimat(e, w)e take a moment to observe that√ 𝑛
𝑛− +𝐴𝐷𝜆 𝑛−13 < 2 𝑛−𝜆 .
This estimate is tedious, but not diffi(cul√ )t. One can show that𝑛
3 𝑛−
+𝐴𝐷
𝜆 𝑛−1
< 2 𝑛−𝜆
occurs if and only if ( )
2 𝑛+𝐴𝜆
2 < √ .
3
Estimating 𝐴 from below by
√︁ 1𝐴 >
(1 − 2(𝑛 + 1)/𝑛2)2
√
and estimating 𝜆 from below by 𝜆 > 2𝑛 g(ives)that
2 𝑛+𝐴𝜆
2 < √
3
61
is implied by ( ) + 𝑛2√𝑛 2𝑛2 𝑛2 √−2𝑛 2𝑛+2+2𝑛+2
2 < √ .
3
Upon observing that √
+ 𝑛
2 2𝑛
𝑛 √ ⩾ 20
𝑛2 − 2𝑛 2𝑛 + 2 + 2𝑛 + 2
when 𝑛 ⩾ 6 for instance, one can now(see )that √
2 𝑛+
𝑛2 2𝑛
𝑛2
√
−2𝑛 2𝑛+2+2𝑛+2
2 < √
3
and as a result, we must have (√ ) 𝑛− +𝐴𝐷𝑛 𝜆 𝑛−13 < 2 𝑛−𝜆 .
We can now conclude
1 𝜆
𝑌0 = (2𝐶) 𝑛−𝜆 (4𝑒𝐴1) 𝑛−𝜆
( ( 1= 2 2𝑛1/2𝑀 (𝐹))𝑛) − (4𝑒𝐴(log 𝑀 (𝐹)+ 𝑛2 )𝑛 𝜆 ) 𝜆𝑛−𝜆
1+𝑛+2𝜆 𝑛
= 2 𝑛−𝜆 · · ( ) 𝑛 𝐴𝐷 (log 𝑀 (𝐹)+ 𝑛𝑛 2(𝑛−𝜆) 𝑀 𝐹 )𝑛−𝜆 𝑒 2
1+𝑛+2𝜆 𝑛· · ( ) 𝑛= 2 − 𝑛 2(𝑛−𝜆) 𝑀 𝐹 − +𝐴𝐷𝑛 𝜆 𝑛 𝜆 · 𝐴𝐷𝑛𝑒 2
1+𝑛+2𝜆 𝑛 √
⩽( 2 )𝑛−𝜆 ·
𝑛 𝐴𝐷𝑛
𝑛 2(𝑛−𝜆) · ( 3𝐻 (𝐹)) 𝑛− +𝐴𝐷𝜆 · 𝑒 2 .
√ 𝑛 +𝐴𝐷
Next we use the fact that 3 𝑛−𝜆 < 2 𝑛−1𝑛−𝜆 to find that
1+𝑛+2𝜆+ 𝑛−1 𝑛 𝑛 𝐴𝐷𝑛𝑌0 < 2 𝑛−𝜆 𝑛− +𝐴𝐷𝜆 · 𝑛 2(𝑛−𝜆) · 𝐻 (𝐹) − +𝐴𝐷𝑛 𝜆 · 𝑒 2
2𝑛+2𝜆
= 2 − +𝐴𝐷 · 𝐷(+1𝑛 𝜆 𝑛 2 ( · 𝐴𝐷𝑛𝐻 (𝐹)1+𝐷+𝐴𝐷)· 𝑒 2 )
( )𝜒 · ((2𝑛 + 2𝜆= 𝐻 𝐹 𝑛 exp − + 𝐷 + 1 𝐴𝐷𝑛𝐴𝐷 log(2)) + log 𝑛 +𝑛 𝜆 2 2 )
= 𝐻 (𝐹)𝜒 4𝜆 + 2(𝑛 − 𝜆)𝑛 · exp ( − + ( ) + 𝐷 + 1 𝐴𝐷𝑛𝐴𝐷 log 2 log 𝑛 +𝑛 𝜆 2 ) 2
𝐷 + 1 𝐴𝐷𝑛
= 𝐻 (𝐹)𝜒𝑛 · exp (4𝐷 + 2 + 𝐴𝐷) log(2) + log 𝑛 +
2 2
= 𝐻 (𝐹)𝜒𝑛 · 𝑒𝜋𝑛
= 𝑌𝐹 .
□
62
Proof of Lemma 2.46. Since 𝑝 is a large special solution, we have
𝑞
𝑝 > 𝑞 ⩾ 𝑌 = 𝐻𝜒𝑛𝑒𝜋𝑛𝐹 .
Since 𝑝 belongs to 𝛼, Thomas’ Corollary 3.1 in [Tho00] indicates that𝑞  𝑝 − 1 1𝛼 < < .𝑞 𝑝𝑛∗𝑞 𝑛/2𝑌
𝐹
if we recall that 𝑛
∗ 𝑛−2
   = 2 by its definition in (2.18). Suppose, by contradiction, that there exists 𝛽 ∈ C with 𝑓 (𝛽) = 0 and𝑝 − 𝑝𝛽 < − 𝛼 . Then by the triangle equality, we find that𝑞 𝑞   𝑝   |𝛼 − 𝛽 | ⩽ − 𝛽 +  𝑝 − 𝛼𝑞 𝑞 
2
< /2 . (2.30)𝑛
𝑌
𝐹
Since 𝛼 and 𝛽 are distinct roots of 𝑓 , Theorem 4 in [Rum79] indicates that
| 1 1𝛼 − 𝛽 | >
2𝑛𝑛/2+2
=
(4𝐻)𝑛 22
. (2.31)
𝑛+1𝑛𝑛/2+2𝐻𝑛
Combining (2.30) and (2.31), we find that
1 2
22𝑛+1𝑛𝑛/2+2
<
𝐻𝑛 𝑛/2𝑌
𝐹
and rearranging yields
𝑛/2
𝑌
𝐹
22𝑛+2𝑛𝑛/2+2
< 1.
𝐻𝑛
From here, we can use the fact that 𝑌𝐹 = 𝐻𝜒𝑛𝑒𝜋𝑛 to find
𝐻𝑛(𝜒𝑛/2−1)𝑒𝑛𝜋𝑛/2
1 >
22𝑛+2𝑛𝑛/2+2
𝑒𝑛𝜋𝑛/2
⩾
22𝑛+2𝑛𝑛/2+2
where the last inequality follows because 𝜒𝑛 ⩾ 2. After rearranging, this implies that
(4𝑛 + 4) log(2) + (𝑛 + 4) log(𝑛)
𝜋𝑛 <
𝑛
⩽ 5 log(2) + 2 log(𝑛)
where the last inequality follows from the fact that 𝑛 ⩾ 6. However, the last inequality
contradicts our hypothesis that 𝜋𝑛 ⩾ 5 log(2) + 2 log(𝑛), so no such 𝛽 can exist and the
closest root of 𝑓 (𝑥) to 𝑝/𝑞 is 𝛼. □
63
Finally, we prove Proposition 2.44.
Proof of Proposition 2.44. Let 𝛼 > 1 be a real root of 𝑓 (𝑥). By Lemma 2.45, every large
special solution (𝑝, 𝑞) so that 𝑝/𝑞 belongs to 𝛼 is large in the sense of [BS87]. Moreover,
by Lemma 2.46, any large special solution (𝑝, 𝑞) so that 𝑝/𝑞 belongs to 𝛼 has 𝑝 −   𝑝𝛼 = min − 𝛽 .𝑞 𝑓 (𝛽)=0 𝑞 
Hence, every large special solution (𝑝, 𝑞) so that 𝑝/𝑞 belongs to 𝛼 is large (in the sense of
[BS87]) and is nearest to 𝛼 among all the roots of 𝑓 (𝑋). Lemma 2 of [BS87] indicates
that there are no more than ⌊ ⌋
log 𝐸 + 2 log(𝑛) − log(𝐿 − 2)
𝑍 = + 2
log(𝑛 − 1)
large solutions (𝑝, 𝑞) so that 𝑝/𝑞 is nearest to 𝛼 among all the roots of 𝑓 (𝑋) and so we
conclude that there are no more than 𝑍 large special solutions (𝑝, 𝑞) with 𝑝/𝑞 belonging
to 𝛼. □
2.7.3 Choosing Parameters for Large Degrees
Begin by assuming 𝑛 ⩾ 507. We handle all smaller instances of 𝑛 computationally.
Recall that 𝑛∗ = 𝑛−22 from its definition in (2.18).
Proposit√︃ion 2.47. ∗For 𝑛 ⩾ 507, we can take 𝑑0 = 𝑛2 , 𝑑 = 𝑛∗, 𝑎 = 14 , 𝐶 = 7/6, 𝑐 = 89𝐶2−1 ,
2𝑛+ 1
= 1 − 8𝑏 2 and obtain 𝑇 = 2 and 𝑍 = 2.𝑐𝑛
𝑛−1+2
Observe first that these are the smallest possible values of 𝑇 and 𝑍 .
Proof. To show this, we first must show that these choices of 𝑑0, 𝑑, 𝑎, and 𝑏 meet the
requirements listed in (2.22), (2.23), (2.24), and (2.26).
Certainly 0 ⩽ 𝑑0 ⩽ min(𝑛∗ − 1.4, 𝑑) and 1 < 𝑑 ⩽ 𝑛∗. All that remains to show for 𝑑0
and 𝑑 is (2.24). We have ( )
𝑐0 𝑑−1
𝑄𝑑−1
𝑝
1 =
0
(𝐾0 𝑛∗
2𝑛∗/2
)
2 −1
⩾ .
𝐾𝑛∗/2(𝑛)
64
But observe that √︄ ( ( √︄ ))𝑛∗/2
2𝑛 2
𝐾𝑛∗/2(𝑛) = 2 ( − () ( − ) 2.032
1/𝑛
) 1 +𝑛 1 𝑛 2 (𝑛 − 2)𝑝𝑛0√︁ 𝑛−2· + 1 4⩽ 2 (2.032 1 ) (𝑛 − 2)2𝑛−1𝑛−2
1 4
⩽ 5 1 +
𝑛 − 2
⩽ 5𝑒1/4
so 𝑄𝑑−11 is certainly greater than 1. Similar reasoning shows that 𝐾𝑑 (𝑛) ⩽ 5𝑒
1/2, so it is
certainly also the case that ( ∗/2 ) 𝑛∗ −1 ( 𝑛∗2𝑛 2 2𝑛∗/2 ) 2 −1
𝐾 (𝑛) ⩽ 5𝑒1/2 < 𝑑−1𝑑 5𝑒1/4
⩽ = 𝑄 .
𝐾 1𝑛∗/2(𝑛)
Hence, our choices of 𝑑 and 𝑑0 are valid.
Next, we wish to check that our choices for 𝑎 and 𝑏 are valid. To check 0 < 𝑎 < 𝑏,
note that √︃
2𝑛 + 1 √ √
− 8 − 2 𝑛𝑏 = 1 2 ⩾ 1 = 1 −
2(𝑛 − 1) 𝑛
⩾ 1 − 2√
𝑐𝑛
−1 + 2
𝑐𝑛2 𝑐𝑛2
𝑛 𝑛− 𝑐 𝑛1
− 2 1⩾ 1 √ > = 𝑎. (2.32)
𝑐 507 4
√
To check that 𝑏 < 1 − 2𝑛+2𝑎2 , it suffices t√o show that
𝑐𝑛2𝑛 > −1 + 2. But this occurs if and𝑛 𝑛
only if (1 − ) 2 − 3 + 2 0, i.e. 3+ 9−8(1−𝑐)𝑐 𝑛 𝑛 > 𝑛 > 2(1− ≈ 9.66, which we certainly have.𝑐)
To show that 𝑇 = 2, we claim that we have the following two inequalities (and from
equation (2.21) together with the fact that 𝑑( −1)+ ⩽ 1, it will follow that 𝑇 = 2):𝑑0 𝑑 𝑑
𝜒𝑛𝑛(𝑑 − 1)
𝑑0(𝑑 − 1) +
+ 1 < 𝑑, (2.33)
𝑑
𝜋𝑛
1 + 1 < 𝑑. (2.34)log𝐾 −𝑑 (𝑛) 𝑑−1𝑄1
We first show (2.33). Sub(stitut)ing
∗
( 𝑑
𝑛
0 = )2 and 𝑑 = 𝑛
∗, observe that (2.33) is
equivalent to
𝑛−2 𝑛−4
4 2 − 1 +
𝑛−2
2 𝑛 − 2 𝑛∗
𝜒𝑛 < = = .
𝑛 8 4
65
Keeping an eye on the definition of 𝜒𝑛 given in equation (2.27), we have that
𝐴 = 16,
7
𝐶 = ,
6
8 32
𝑐 = = ,
9𝐶2
(√︃
− 1 45
2𝑛 + 1
− 8𝑏 = 1 ) .
𝑐𝑛2
−1 + 2𝑛
All of these together yield
𝑐𝑛2
( )
+ 32 1𝐿 = − 2 = 𝑛 + 1 + − + 2 (2.35)𝑛 1 45 𝑛 1
and it is now easy to check that
32 32
𝑛 ⩽ 𝐿 ⩽ 𝑛 + 3.
45 45
From here we have
3
𝐿
= ⩽ 4𝐷 − (25𝑛 + 3 ) 3245𝑛 + 3 32 6075=𝑛 𝐿 − 32 + 13 = + ⩽ 2.54𝑛 𝑛 3 45𝑛 − 3 13 13(13𝑛 − 135)45
32
⩾ 45
𝑛 32
𝐷 32 = ≈ 2.46𝑛 − 45𝑛 13
when we use the fact that 𝑛 ⩾ 507. To convert these into estimates on 𝜒𝑛, we have
94853
𝜒𝑛 = 17𝐷 + 1 ⩽ ⩽ 44.08, (2.36)2152
+ 557𝜒𝑛 = 17𝐷 1 ⩾ ⩾ 42.8. (2.37)13
Since 𝑛 ⩾ 507, we now have 𝜒𝑛 ⩽ 44.08 < 𝑛−28 which confirms equation (2.33).
Equation (2.34) is more complicated to handle. Observe that by equation (2.28), we
have
( ( + ) + ) + (𝐷 + 1) log 𝑛 + 𝐴𝐷𝑛𝜋𝑛 = 𝐷 4 𝐴 2 log 2 2 2
⩽ 36.6 + 1.77 log 𝑛 + 20.28𝑛
⩽ 37 + 21𝑛. (2.38)
66
For reference later, we will also note
𝜋𝑛 ⩾ 35.5 + 1.7 log 𝑛 + 19.6𝑛
⩾ 46 + 19𝑛. (2.39)
It will additionally be help√︄ful for us to have (an estimat(e on√︄𝐾𝑑 (𝑛). We h)a)ve𝑑
( ) 2𝑛 2𝐾𝑑 𝑛 = 2√︄( − )( − ) (2.032
1/𝑛 1 +
𝑛 1 𝑛 2 √︄ ()𝑛 − 2)𝑝𝑛0𝑑
3𝑛 − 3 + 2𝑛 − 4⩽ 4√︂ (𝑛 − 1() (𝑛 −√︄ 12) ) (𝑛 − 2)𝑝𝑛0𝑛∗
3 2
⩽ 4 1 + .
𝑛 − 2 𝑝𝑛0
Now, since 𝑝0 ⩾ 2, we ha√︂ve ( √︂ ) 𝑛−2 √︂
2 ( ) 𝑛−23 2 3 1 2
𝐾𝑑 (𝑛) ⩽ 4 − 1 + ⩽ 4 − 1 +𝑛 2 2𝑛√︂ 𝑛 2 2
𝑛−2
2
3 © 1
⩽ 4 ­«­1 + ( ) ®
ª 𝑛−2® 2 √︂ 3− ¬ ⩽ 4𝑒 (2.40)𝑛 2 𝑛−2 𝑛 − 22
and similar reasoning yields √︂ ( )
3 1 𝑛
∗/2
𝐾𝑑0 (𝑛) ⩽ 4 𝑛 − 1 + ∗ . (2.41)2 2𝑛
Combining the upper bounds in equation (2.38) with the fact that for 𝑛 ⩾ 270,
37 + log 1.921𝑛 < (𝑛 − 2) (𝑛 − 4)
8
yields
log(1.9)
𝜋𝑛 < (𝑛 − 2) (𝑛 − 4).8
Now ine[qualities (2.40]) and (2[.41) give
( − ) ( 1
]
𝑑 1 log 𝐾𝑑 𝑛)− −1𝑄 = log 𝐾 (𝑛)−1𝑄(𝑑−1𝑑 1  𝑑 1 ) ( )1/2 𝑐0 𝑛∗−11 𝑝 
⩾ log  · 𝑒−1 ·
𝑛 − 2 · 0 
4 3 𝐾 0 
67
[ ( )1/2 ( ∗/2 ) ∗−1]1 𝑛 − 2 2𝑛 𝑛
⩾ log · 𝑒−1 ·
4 ( 3 ) (𝐾𝑑0 (𝑛) ) 𝑛−4 ( √︂ ) 𝑛−41/2 ∗/2 2 2 1 𝑛
⩾ log [ · 𝑒
−1 · 𝑛 − 2 2 1 𝑛 − 2 
(4 ) 3 ( + ∗1 2−𝑛∗)𝑛 /2 4 3 𝑛−2 ( ) 𝑛−2 ( ) ( 𝑛−2 ) ( 𝑛−4 ]1 2 𝑛 − 2 4 − 2 4 2 )⩾ log [ 1( 𝑒4 ) 3 1 + 2−𝑛∗𝑛−2 ]
𝑛 − 2 4 (𝑛−2) (𝑛−4)
⩾ log 𝑒−1 · 1.9 8
48 ( )
log 1.9 ( − )( − 𝑛 − 2 𝑛 − 2⩾ 𝑛 2 𝑛 4) + log − 1
8 4 48
log(1.9)
⩾ (𝑛 − 2) (𝑛 − 4)
8
> 𝜋𝑛
which now implies that equation (2.34) is satisfied. Hence, we conclude that 𝑇 = 2.
Finally, we check that 𝑍 = 2. In order to use Proposition 2.44, we must verify that
𝜒𝑛 ⩾ 2 and 𝜋𝑛 ⩾ 5 log(2) + 2 log(𝑛). However, these quickly follow from (2.37) and
(2.39).
As before in equation (2.32), we have 𝑏 ⩾ 1 − √2 > 0.87509, so
𝑐 507
1
𝐸 = < 0.711 < 𝑐
2(𝑏2 − 𝑎2)
and so (also using (2.35)) ( )
log 𝑐𝑛2log 𝐸 + 2 log(𝑛) − log(𝐿 − 2) 𝐿−2
log(𝑛 − ) < = 1.1 log(𝑛 − 1)
Therefore, by (2.29), we note that 𝑍 = 2. □
Note that Proposition 2.47 proves Lemma 2.40 for 𝑛 ⩾ 507.
2.7.4 Choosing Parameters for Small Degrees
For 𝑛 ⩽ 506, we make parameter choices listed in table A.1. One can check that the
parameter choices satisfy (2.24), (2.26), (2.22), and (2.23) along with the necessary
bounds on 𝜋𝑛 and 𝜒𝑛 in order to use Proposition 2.44, and yield the 𝑇 and 𝑍 values giving
68
𝑧(𝑛) = 𝑇 + 𝑍 + 1 when 6 ⩽ 𝑛 ⩽ 8 and 𝑧(𝑛) = 𝑇 + 𝑍 for 𝑛 ⩾ 9. A Jupyter notebook, whose
code is contained in appendix B.1 produced these parameters and verified that these
parameters are valid and yield the conclusion of Lemma 2.40 for 𝑛 ⩽ 506, which
concludes our investigation.
In brief, the code picks a value of 𝑛, sets 𝑑 = 𝑛∗, brute force loops over a number of
valid values for the parameters 𝑑0, 𝑎, 𝑏, computes the corresponding 𝑇 and 𝑍 values
defined in equations (2.21) and (2.29), and records the values of 𝑑0, 𝑎, and 𝑏 which
minimize 𝑇 + 𝑍 . The following table contains some of the more interesting data points in
table A.1.
𝑛 𝑑0 𝑑 𝑎 𝑏 𝑇 𝑍
6 0 2 0.18 0.29 10 4
7 0.539 2.5 0.2 0.28 7 4
8 0.992 3 0.16 0.41 7 3
9 0.882 3.5 0.17 0.4 6 3
10 1.17 4 0.23 0.41 5 3
11 1.674 4.5 0.14 0.37 5 3
12 2.088 5 0.27 0.41 4 3
13 2.255 5.5 0.2 0.37 4 3
14 2.484 6 0.16 0.35 4 3
15 2.958 6.5 0.13 0.34 4 3
16 3.136 7 0.11 0.32 4 3
17 3.904 7.5 0.32 0.42 3 3
18 4.158 8 0.27 0.39 3 3
.. . . . . . .. .. .. .. .. .. ..
36 8.268 17 0.08 0.26 3 3
37 7.728 17.5 0.08 0.25 3 3
38 11.454 18 0.44 0.48 2 3
39 11.799 18.5 0.4 0.45 2 3
.. .. .. .. . . .
. .
. .. .
.
. ..
215 59.907 106.5 0.03 0.87 3 2
216 59.136 107 0.03 0.87 3 2
217 67.3735 107.5 0.389816 0.881816 2 2
218 67.9042 108 0.399038 0.883038 2 2
.. .. .
.
. .
. . . .
. .. .. .. ..
69
𝑛 𝑑0 𝑑 𝑎 𝑏 𝑇 𝑍
506 218.022 252 0.517076 0.927076 2 2
507 218.457 252.5 0.517138 0.927138 2 2
Table 2.1. Summary of parameter choices which minimize 𝑇 + 𝑍
2.7.5 Attaining Bounds With Examples
Theorem 1.17 indicates that for 𝑛 ⩾ 217, there are no more than 40 distinct solutions
to equation (2.4) when 𝐹 (𝑥, 𝑦) is a trinomial. For smaller 𝑛, this upper bound is even
larger. Of interest is whether or not it is possible to find a particular trinomial for which
(2.4) has 40 distinct solutions. The computer algebra system GP has a method called thue
which, on input a Thue equation, will output the solutions to that Thue equation‡. The
author has used this method to create a function in Sage which, on input a degree 𝑛 and
height 𝐻, will compute the solutions to every trinomial Thue equation of degree 𝑛 and
height 𝐻. The method can be found in appendix B.1 and the raw data can be found on the
author’s website at:
https://pages.uoregon.edu/gknapp4/files/trinomial_solution_data.zip
The maximal number of solutions to equation (2.4) for a trinomial 𝐹 (𝑥, 𝑦) of degree 𝑛
and height 𝐻 are found in the two tables below. Notably, no trinomial has been found with
more than 12 solutions to (2.4), which is far from the upper bound of 40. Moreover, while
much of the data in the tables give the notion that the maximal number of solutions only
depends on 𝐻, the column 𝐻 = 16 confirms that the data supporting such a hypothesis is
coincidental. A hyphen in the table means that the case in question has not yet been
computed.
‡While the accuracy ofGP’sthuemethod relies on theGeneralizedRiemannHypothesis
to solve the Thue equation 𝐹 (𝑥, 𝑦) = ℎ, our use of it does not because thue does not assume
GRH when solving the specific equation 𝐹 (𝑥, 𝑦) = ±1.
70
𝐻 = 1 𝐻 = 2 𝐻 = 3 𝐻 = 4 𝐻 = 5 𝐻 = 6 𝐻 = 7 𝐻 = 8 𝐻 = 9
𝑛 = 6 8 6 8 8 6 6 6 6 8
𝑛 = 7 8 6 8 8 6 6 6 6 8
𝑛 = 8 8 6 8 8 6 6 6 6 8
𝑛 = 9 8 6 8 8 6 6 6 6 8
𝑛 = 10 8 6 8 8 6 6 6 6 8
𝑛 = 11 8 6 8 8 6 6 6 6 8
𝑛 = 12 8 6 8 8 6 6 6 - -
𝑛 = 13 8 6 8 8 6 6 - - -
𝑛 = 14 8 6 8 8 6 6 - - -
𝑛 = 15 8 6 8 8 6 - - - -
𝑛 = 16 8 6 8 8 6 - - - -
𝑛 = 17 8 6 8 8 - - - - -
𝑛 = 19 8 6 8 - - - - - -
𝑛 = 20 8 6 8 - - - - - -
Table 2.2. The maximal number of solutions to equation (2.4) for a trinomial 𝐹 (𝑥, 𝑦) of
degree 𝑛 and height 𝐻 ⩽ 9.
𝐻 = 10 𝐻 = 11 𝐻 = 12 𝐻 = 13 𝐻 = 14 𝐻 = 15 𝐻 = 16 𝐻 = 17
𝑛 = 6 6 6 6 6 - 6 12 6
𝑛 = 7 6 6 6 6 6 6 8 6
𝑛 = 8 6 6 6 6 6 - 12 -
𝑛 = 9 6 6 6 - - - 8 -
𝑛 = 10 - 6 - - - - - -
Table 2.3. The maximal number of solutions to equation (2.4) for a trinomial 𝐹 (𝑥, 𝑦) of
degree 𝑛 and height 𝐻 ⩾ 10.
71
CHAPTER 3
ROOT SEPARATION
3.1 Introduction
Recall that we used Rump’s result (2.31) in our proof of Lemma 2.46. Rump’s result
bounding the distances between distinct roots of a given polynomial is not just useful as a
technical tool, but also is part of a well-studied tradition regarding the geometry of roots of
polynomials. This tradition includes Descartes’ Rule of Signs, the Gauss-Lucas Theorem,
and the Schinzel-Zassenhaus Conjecture, to name a few key results. Bounds on root
separation in particular have computational applications to root-finding algorithms as
Koiran outlines in [Koi19]. In this chapter, we focus primarily on bounding the separation
between roots of a fixed polynomial. Recall the definition of a polynomial’s separation,
given in Definition 1.25 and restated here for convenience.
Definition. Given a polynomial 𝑓 (𝑥) ∈ C[𝑥] with roots 𝛼1, . . . , 𝛼𝑛 ∈ C, the separation of
𝑓 (𝑥) is the quantity
sep( 𝑓 ) = min |𝛼𝑖 − 𝛼 𝑗 |.
𝛼𝑖≠𝛼 𝑗
In particular, we look at upper bounds on the separation of a polynomial. Our main
conjecture is Conjecture 1.33, which is reprinted here for convenience:
Conjecture. Suppose 𝑓 (𝑥) ∈ R[𝑥] is monic and separable of degree 𝑛 ⩾ 2. If 𝑓 (𝑥) has
any real roots, then
sep( 𝑓 ) ≪𝑛 𝑀 ( 𝑓 )1/(𝑛−1) .
If 𝑓 (𝑥) has only nonreal roots, then
sep( 𝑓 ) ≪𝑛 𝑀 ( 𝑓 )1/𝑛.
In the course of this chapter, we prove Theorem 1.34, which we restate here for
convenience.
Theorem. Let 𝑓 (𝑥) ∈ R[𝑥] be monic and separable with deg( 𝑓 ) = 𝑛 ⩾ 2 and suppose
that any of the following conditions is met.
1. deg( 𝑓 ) = 2.
2. deg( 𝑓 ) = 3.
72
3. deg( 𝑓 ) = 4 and 𝑓 (𝑥) has no real roots.
4. Every root of 𝑓 (𝑥) is real.
Then if 𝑓 (𝑥) has any real roots,
sep( 𝑓 ) ≪ 𝑀 ( 𝑓 )1/(𝑛−1)𝑛 .
If 𝑓 (𝑥) has only nonreal roots, then
sep( 𝑓 ) ≪𝑛 𝑀 ( 𝑓 )1/𝑛.
In fact, we prove something more precise: Propositions 3.7 through 3.10 give
(sometimes sharp) explicit bounds on sep( 𝑓 ) for the cases given in Theorem 1.34.
3.2 Context
3.2.1 Lower Bounds on Separation
The starting point for much research around bounds on separation is Corollary 1.32,
stated again here for convenience.
Theorem (Mahler). Suppose that 𝑓 (𝑥) ∈ Z[𝑥] is separable of degree 𝑛 ⩾ 2. Then
√
3
sep( 𝑓 ) > .
𝑛(𝑛+2)/2𝑀 ( 𝑓 )𝑛−1
Mahler’s theorem gives an important lower bound for separation in terms of Mahler
measure. Many others have developed this theory further. For example, Rump removed
the separability hypothesis when he proved in [Rum79] that
Theorem 3.1 (Rump). Suppose that 𝑓 (𝑥) ∈ Z[𝑥] has degree 𝑛 and let 𝑆 denote the sum of
the absolute values of the coefficients of 𝑓 (𝑥). Then
1
sep( 𝑓 ) >
2𝑛𝑛/2+2
.
(𝑆 + 1)𝑛
Since 𝑆 ⩽ 2𝑛𝑀 ( 𝑓 ), this yields the relation
1
sep( 𝑓 ) >
2𝑛𝑛/2+2(2𝑛𝑀 ( 𝑓 ) + 1)𝑛
which is numerically worse than Mahler’s theorem, but does encompass more cases.
Others—especially Bugeaud, Dujella, Pejković—have focused on improving the
numerics of Mahler’s theorem. To examine their results, we introduce some new notation.
73
Notation 3.2. Let 𝑓 (𝑥) ∈ C[𝑥] be a polynomial. Then let
( ) − log sep( 𝑓 )𝑒 𝑓 :=
log 𝑀 ( 𝑓 ) (3.1)
so that 𝑒( 𝑓 ) always satisfies the relation
1
sep( 𝑓 ) = .
𝑀 ( 𝑓 )𝑒( 𝑓 )
Then, for an integer 𝑛 ⩾ 2, let
𝑒(𝑛) := sup{𝑒( 𝑓 ) : 𝑓 (𝑥) ∈ Z[𝑥], deg( 𝑓 ) = 𝑛, 𝑓 is separable},
𝑒irr(𝑛) := sup{𝑒( 𝑓 ) : 𝑓 (𝑥) ∈ Z[𝑥], deg( 𝑓 ) = 𝑛, 𝑓 is irreducible},
𝑒∗red(𝑛) := sup{𝑒( 𝑓 ) : 𝑓 (𝑥) ∈ Z[𝑥], deg( 𝑓 ) = 𝑛, 𝑓 is reducible and monic}
Now, 𝑒(𝑛) represents the minimal value of 𝑒 which makes the following statement
true: there exists a constant 𝐶 (𝑛) so that for every separable 𝑓 (𝑥) ∈ Z[𝑥] of degree 𝑛,
( ) 𝐶 (𝑛)sep 𝑓 >
𝑀 ( 𝑓 ) .𝑒
The quantity 𝑒irr(𝑛) plays the same role, but for the more restrictive class of irreducible
polynomials, so 𝑒irr(𝑛) ⩽ 𝑒(𝑛) for all 𝑛. Mahler’s theorem then implies that 𝑒(𝑛) ⩽ 𝑛 − 1,
but it remains possible that 𝑒(𝑛) could be smaller. Likewise, Theorem 3.1 implies that
𝑒∗red(𝑛) ⩽ 𝑛.
By constructing families of examples, Bugeaud and Dujella in [BD11] showed that
𝑛 𝑛 − 2
𝑒irr(𝑛) ⩾ +2 4(𝑛 − 1) ,
demonstrating that 𝑛 is at least the right order of magnitude for 𝑒(𝑛) and 𝑒irr(𝑛). On the
other hand, Dujella and Pejković in [DP17] showed that
𝑒∗red(𝑛) ⩽ 𝑛 − 2,
which indicates that Rump’s Theorem 3.1 can very likely be improved.
Of course, the degree is not the only quantity which impacts the separation of a
polynomial. The number of nonzero summands of 𝑓 (𝑥) also impacts the separation, as we
note that Example 1.30 shows that for m(onic bi(nom)i)als 𝑓 (𝑥),
sep( 𝑓 ) 𝜋 1> 2 +𝑂 3 𝑀 ( 𝑓 )
1/𝑛.
𝑛 𝑛
Koiran, in [Koi19], uses Baker’s bounds on linear forms in logarithms to show:
74
Theorem 3.3 (Koiran). Suppose 𝑓 (𝑥) ∈ Z[𝑥] is a trinomial and let
𝑌 = log max(𝐻 ( 𝑓 ), deg( 𝑓 )).
Then for some absolute constant 𝐶,
sep( 𝑓 ) > exp(−𝐶𝑌3).
Each of these results has thus far explored lower bounds on the separation in terms of
the Mahler measure. In the next section, we turn the question around and ask about upper
bounds on separation in terms of the Mahler measure.
3.2.2 Upper Bounds on Separation
In the introduction to this dissertation, we noted that there is a trivial upper bound on
separation given by the Mahler measure via inequality (1.7). This can be improved upon
without too much work from a surprising source. Theorem 1.31 gives a lower bound on
separation prima facie, but can be manipulated to produce an upper bound on separation.
Proposition 3.4. Suppose that 𝑓 (𝑥) ∈ C[𝑥] is separable of degree 𝑛 ⩾ 2 and leading
coefficient 𝑏 ≠ 0. Then
𝑛+2 ( ) 2(𝑛−1)
𝑛 𝑛2−𝑛−2 𝑀 ( 𝑓 ) 𝑛2−𝑛−2
sep( 𝑓 ) < · .
31/(𝑛2−𝑛−2) |𝑏 |
If 𝑛 ⩾ 4, then ( )
( 2/(𝑛−
1 )
( ) 1 𝑀 𝑓 )
2
sep 𝑓 < 𝑛 𝑛−3 | .𝑏 |
Proof. Observe first that since 𝑓 (𝑥) is separab∏le, we have
|Δ 𝑓 | = |𝑏 |2𝑛−2 |𝛼 2𝑖 − 𝛼 𝑗 |
1⩽𝑖< 𝑗⩽𝑛
⩾ |𝑏 |2𝑛−2 2sep( 𝑓 )𝑛 −𝑛.
We can apply this fact along with Theo√︁rem 1.31 to find that
3|Δ |
sep( 𝑓𝑓 ) >
𝑛(𝑛+2)/2√ 𝑀 ( 𝑓 )
𝑛−1
3|𝑏 |𝑛−1 sep( 2𝑓 ) (𝑛 −𝑛)/2
⩾
𝑛(𝑛+2)/2𝑀 ( 𝑓 )𝑛−1
75
and rearranging yields
𝑛+2 ( ) 2(𝑛−1)
𝑛 𝑛2−𝑛−2( ) · 𝑀 ( 𝑓 )
𝑛2−𝑛−2
sep 𝑓 < .
31/(𝑛2−𝑛−2) |𝑏 |
If 𝑛 ⩾ 4, it is easy to verify that 𝑛+2 12− −2 ⩽ −3 and
2(𝑛−1) 2
2− −2 ⩽ − 1 , which concludes the𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 2
proof. □
Observe that Proposition 3.4 gives a much better bound that we found in (1.7).
Moreover, Proposition 3.4 lends more credence to the philosophy that “polynomial
roots are not randomly distributed.” Vaguely, what we mean by this is that while the roots
of 𝑓 (𝑥) must lie in the complex disk of radius 1 + 𝐻 ( 𝑓 )| | , they are not uniformly distributed𝑏
within that disk. If they were, separation would satisfy a much different bound. Here, we
examine the expected value of the minimum separation between two points when 𝑛
random points are selected inside a disk of specified radius. We follow a similar line of
reasoning that can be found in Hernan Gonzalaz’ StackExchange answer given at the URL
in the footnotes.∗
Proposition 3.5. Suppose that 𝑛 points 𝑆 = {𝛼1, . . . , 𝛼𝑛} are scattered independently with
uniform probability in the complex disk of radius 𝑅 > 0, centered at 0. Then the expected
value of sep(𝑆) := min𝑖≠ 𝑗 |𝛼𝑖 − 𝛼 𝑗 | is at least
√
2 2 · 𝑅 .
3 𝑛
Proof. Let 𝑝𝑛 (𝑥) denote the probability th∫at sep(𝑆) ⩾ 𝑥. Then the expected separation is∞
𝐸𝑛,𝑅 := 𝑝𝑛 (𝑥) 𝑑𝑥.
0
We will place 𝑛 balls with diameter 𝑥 randomly uniformly inside 𝐵𝑅 (0). Say that
those balls are 𝐵1, . . . , 𝐵𝑛 which have respective centers 𝛼1, . . . , 𝛼𝑛. For every
𝑗 ∈ {{𝑖, 𝑘} : 1 ⩽ 𝑖 < 𝑘 ⩽ 𝑛}, let 𝑆 𝑗 = 𝑆𝑖,𝑘 denote the event that 𝐵𝑖 ∩ 𝐵𝑘 ≠ ∅. Now, 𝑃(𝑆 𝑗 )
is equal to the probability that 𝛼𝑘 is placed within distance 𝑥 of 𝛼𝑖, which is at most
𝜋𝑥2 𝑥2
( )
2 = 2 . Hence, 𝑃(𝑆 𝑗 ) ⩽ 𝑥
2.
𝜋𝑅 𝑅 𝑅
Then we note that
𝑝𝑛 (𝑥) = 𝑃(∩ 𝑆c𝑗 𝑗 )
∗https://math.stackexchange.com/questions/2005775/average-minimum-distance-
between-n-points-generate-i-i-d-uniformly-in-the-bal
76
= 1 − 𝑃∑︁(∪ 𝑗𝑆 𝑗 )
⩾ 1 − ( 𝑃(𝑆 𝑗 )𝑗 ( )
𝑛 ( 𝑥 ) )2
⩾ max 1 − , 0 .
2 𝑅
Therefore, ∫ ∞
𝐸𝑛,𝑅 = ∫ 𝑝𝑛 (𝑥) 𝑑𝑥0 √/ (𝑛) ( )𝑅 2
− 𝑛
( 𝑥 )2
⩾ 1 𝑑𝑥
2 𝑅
√︁0 √2 2𝑅=
3 𝑛(𝑛 − 1)
√
2 2 · 𝑅⩾ .
3 𝑛
□
Hence, if the roots of monic 𝑓 (𝑥) ∈ C[𝑥] of degree 𝑛 were randomly uniformly
distribute√d in the complex ball of radius 1 + 𝐻 ( 𝑓 ), we would expect their separation to be
at least 2 2 · 1+𝐻 ( 𝑓 )3 . However, Proposition 3.4 shows this not to be the case: the separation𝑛
is always much smaller for polynomials of large height.
The main question that we explore is whether or not Proposition 3.4 can be improved.
Like those who study lower bounds on separation, we largely focus on the exponent of
𝑀 ( 𝑓 ) in the relation sep( 𝑓 ) < 𝐶 (𝑛)𝑀 ( 𝑓 )𝑒. This will give us a quantification for how
nonuniform the distribution of polynomial roots is.
3.3 Data and Conjectures
For the remainder of this chapter, we will examine polynomials with real coefficients,
keeping in mind the fact that our ultimate goal is to understand polynomials with integer
coefficients. We are searching for upper bounds on separation in terms of the Mahler
measure and we aim to produce those bounds by selecting a large number of “random”
polynomials with a certain set of characteristics, plotting the separations against the
Mahler measures of those polynomials, then examining the data to see what region of the
plane these points can lie in.
One of the major challenges with this approach is determining what we mean by a
“random” polynomial. For polynomials in R[𝑥], one could choose a polynomial of degree
77
𝑛 by choosing a random vector from a box in R𝑛+1 to use as the coefficients. Alternatively,
one could select a polynomial of degree 𝑛 by choosing a random element of the star-body
of polynomials of degree 𝑛 with Mahler measure at most 1 (or some other bound), as
Sinclair and Yattselev do in [SY15]. Here, we choose to select a random polynomial by
choosing its roots uniformly from a specified compact subset of the complex plane. The
major advantage of this approach is the speed with which we can compute Mahler measure
and separation. Were we to select the coefficients of the random polynomial, we would
first have to find the roots of the polynomial before we could compute the separation and
Mahler measure, a notoriously difficult problem.
To produce this data, we wrote Sage code which is fully specified and documented in
Appendix B.3. The most important method in that Appendix is PlotMahlerVSep which
takes as input a number of trials 𝑁 , a number of roots 𝑛, a “radius” 𝑅, a number of real
roots 𝑟 (which defaults to 𝑟 = 0), a discriminant lower bound 𝑑 (which defaults to 𝑑 = 0),
and a string indicating the region of the complex plane from which the roots will be
chosen. The method then creates an empty plot 𝑃 and conducts the following experiment
𝑁 times:
1. Select 𝑟 random elements from the specified region of R uniformly and select 𝑛−𝑟2
random elements and their complex conjugates from the specified region of C
uniformly.
2. Set 𝑓 (𝑥) to be the monic polynomial with those 𝑛 elements as roots. If |Δ 𝑓 | ⩾ 𝑑,
proceed to the next step. Otherwise, return to the previous step.
3. Compute the separation and Mahler measure of 𝑓 (𝑥) and add the point
(𝑀 ( 𝑓 ), sep( 𝑓 )) to the plot 𝑃.
If the region is selected to be a box, the real roots are chosen from the interval [−𝑅, 𝑅]
while the complex roots are chosen from the box max ( |ℜ[𝑧] |, |ℑ[𝑧] |) ⩽ 𝑅. If the region
is selected to be a ball, the real roots are chosen from the interval [−𝑅, 𝑅], while the
complex roots are chosen from the ball |𝑧 | ⩽ 𝑅. If the region is selected to be an annulus,
the real roots are chosen from the set [−𝑅,−1/𝑅] ∪ [1/𝑅, 𝑅] while the complex roots are
chosen from the region 1/𝑅 ⩽ |𝑧 | ⩽ 𝑅.
The user is able to specify an annular region because Sinclair and Yattselev show in
[SY15] that a “typical” polynomial has roots clustering around the unit circle in C. This
allows us to potentially spot differences between data sets for typical and atypical
78
polynomials. However, since we want to prove results for all polynomials, we typically
select our region to be a box or a ball.
Finally, we note that since we are interested in identifying the optimal exponent 𝑒 in a
relation like sep( 𝑓 ) < 𝐶 (𝑛) · 𝑀 ( 𝑓 )𝑒, we often find it helpful to plot separation against
Mahler measure on log-log axes.
To identify the most general patterns available, we begin with the least restrictive
computations in the least expensive cases. Figure 3.1 comes from the command
PlotMahlerVSep(50000,2,2,numRealRoots=2),
which selects 50,000 quadratic polynomials whose roots lie in the real interval [−2, 2] and
plots their separations against their Mahler measures:
Figure 3.1. Separation against Mahler measure for monic quadratic polynomials with two
real roots
In contrast, Figure 3.2 comes from the command
PlotMahlerVSep(50000,2,2,numRealRoots=0),
which selects 50,000 quadratic polynomials whose roots are distinct complex conjugates
which lie in the complex ball of radius 2, and plots their separations against their Mahler
measures.
Immediately, we can see the role that the presence of real roots plays in determining
the relation between separation and Mahler measure. Some of these differences are caused
by the anomalies present in the degree two case: for instance, the “empty” space in Figure
3.1 comes from the fact that attaining a Mahler measure near four for a degree two
polynomial requires both roots to have absolute value approximately two, so the two roots
must be either very close together (both near 2 or both near −2) or very far apart (one root
79
Figure 3.2. Separation against Mahler measure for monic quadratic polynomials with no
real roots
near 2 and the other near −2). If we al√lowed larger roots, no such “empty space” would
appear: the polynomial 𝑔(𝑥) = 𝑥2 − 2 5𝑥 + 4 has 𝑀 (𝑔) = 4 and sep(𝑔) = 2, for instance.
However, the sense that the real roots impact the relationship between separation and
Mahler measure is borne out in further examples. Consider the degree three case, where a
polynomial in R[𝑥] can either have one real root or three real roots. Figure 3.3 displays the
results of the command
PlotMahlerVSep(50000,3,2,numRealRoots=3),
which selected 50,000 cubic polynomials with three real roots coming from the interval
[−2, 2]. In some contrast, Figure 3.4 displays the results of the command
PlotMahlerVSep(50000,3,2,numRealRoots=1),
which selected 50,000 cubic polynomials with one real root from the interval [−2, 2] and a
single pair of complex conjugate roots from the complex ball |𝑧 | ⩽ 1.
Some of the differing appearance on these two plots comes from the different scaling
factors on the axes and some is due to a weaker version of the “empty space effect” that we
saw in the degree two case, but there are meaningful differences between the graphs that
are easy to spot: one of those is that the curve which bounds the blue region from above
appears to pass through the point (1, 1) in Figure 3.3 in contrast to the point (1.7, 1) in
Figure 3.4. In fact, this is because a polynomial 𝑓 (𝑥) with 𝑀 ( 𝑓 ) = 1 must have all of its
roots satisfying |𝛼 | ⩽ 1. The maximum separation of the roots of a cubic polynomial if all
three roots are real is then 1 (if the roots are located at −1, 0, 1). On the other hand, if one
80
Figure 3.3. Separation against Mahler measure for monic cubic polynomials with three real
roots
Figure 3.4. Separation against Mahler measure for monic cubic polynomials with one real
root
√
root is real and two are complex, the maximum separation is then 2 sin(𝜋/3) = 3 when
the roots are the three cube roots of unity.
Hence, if we are trying to understand the relation sep( 𝑓 ) < 𝐶 (𝑛)𝑀 ( 𝑓 )𝑒(𝑛) , we should
aim to incorporate differences for polynomials with different signatures.
Definition 3.6. For a polynomial 𝑓 (𝑥) ∈ R[𝑥], the signature of 𝑓 (𝑥) is the pair (𝑟, 𝑠) if
𝑓 (𝑥) has exactly 𝑟 roots in R and 𝑠 pairs of distinct complex conjugate roots.
Moreover, we should begin using log-log plots to illustrate our data if we specifically
want to examine the relation sep( 𝑓 ) < 𝐶 (𝑛)𝑀 ( 𝑓 )𝑒(𝑛) since on a log-log plot, this bound is
linear with slope 𝑒(𝑛):
log sep( 𝑓 ) < log𝐶 (𝑛) + 𝑒(𝑛) log 𝑀 ( 𝑓 ).
81
In fact, this is the exact behavior we see when plotting data on log-log axes. Figure 3.5
is the result of the same operation as Figure 3.2, but plotted on log-log axes and Figure 3.6
is the result of the same operation as Figure 3.1, but plotted on log-log axes.
Figure 3.5. Logarithmic separation against logarithmicMahler measure for monic quadratic
polynomials with no real roots
Figure 3.6. Logarithmic separation against logarithmicMahler measure for monic quadratic
polynomials with two real roots
As expected, the upper bounds on these two regions appears to be linear. More than
that, the upper bound on the blue region in Figure 3.5 appears to approximately be the line
𝑦 = 𝑥 + 0.7 and the upper bound on the blue region in Figure 3.6 appears to approximately
be the line 𝑦 = 𝑥2 + 0.6. Because this is the relatively simple quadratic case, we are able to
precisely determine these bounds later and we do so later in Proposition 3.7.
Continuing to the cubic case, we again see that when plotted on log-log axes, the
upper bound on the blue region appears linear. Figure 3.7 recreates the data of Figure 3.3
on log-log axes and Figure 3.8 recreates the data of Figure 3.4 on log-log axes.
82
Figure 3.7. Logarithmic separation against logarithmic Mahler measure for monic cubic
polynomials with three real roots
Figure 3.8. Logarithmic separation against logarithmic Mahler measure for monic cubic
polynomials with one real root
Very loose estimates of the upper bounds for the blue regions in these figures indicate
that the blue region in Figure 3.7 is bounded above by the line 𝑦 = 𝑥/2 and the blue region
in Figure 3.8 is bounded above by the line 𝑦 = 𝑥/2 + 0.5.
We continue to the quartic case to finish identifying the pattern. Figure 3.9 shows the
results of selecting 100,000 polynomials of degree 4 with four real roots in the interval
[−10, 10] and plotting their logarithmic separations against their logarithmic Mahler
measures. Figure 3.10 does the same thing, but for polynomials with two real roots in the
interval [−10, 10] and a single pair of complex conjugate roots in the ball |𝑧 | ⩽ 10. Figure
3.11 does the same, but for polynomials with two pairs of complex conjugate roots in the
ball |𝑧 | ⩽ 10.
Again eyeballing the upper bounds of the blue region, it appears that the blue region in
Figure 3.9 is bounded above by the line 𝑦 = 𝑥/3 − 1/6, the blue region in Figure 3.10 is
bounded above by the line 𝑦 = 𝑥/3 + 1/6, yet the blue region in Figure 3.11 is bounded
83
Figure 3.9. Logarithmic separation against logarithmic Mahler measure for monic quartic
polynomials with four real roots
Figure 3.10. Logarithmic separation against logarithmic Mahler measure for monic quartic
polynomials with two real roots
Figure 3.11. Logarithmic separation against logarithmic Mahler measure for monic quartic
polynomials with no real roots
above by the line 𝑦 = 𝑥/4 + 1/3.
Since the slope of the linear upper bound corresponds to the exponent 𝑒(𝑛) in the
relation sep( 𝑓 ) < 𝐶 (𝑛)𝑀 ( 𝑓 )𝑒(𝑛) , we are now able to see the data that led to (and supports)
84
Conjecture 1.33. This conjecture is true in certain cases and we verify this in our proof of
Theorem 1.34.
3.4 Proof of Theorem 1.34
Our main goal of this section is to prove Theorem 1.34, which we will do in pieces.
We will moreover give specific constants for various degrees and signatures of 𝑓 (𝑥) in the
inequality sep( 𝑓 ) ≪𝑛 𝑀 ( 𝑓 )1/(𝑛−1) . We prove the following propositions:
Proposition 3.7. Let 𝑓 (𝑥) ∈ R[𝑥] have degre(e 2 and) leading coefficient 𝑏. If 𝑓 (𝑥) has noreal roots, then
( ) 1/2𝑀 𝑓
sep( 𝑓 ) ⩽ 2 | .𝑏 |
If 𝑓 (𝑥) has two real roots, then ( )
𝑀 ( 𝑓 )
sep( 𝑓 ) ⩽ | | + 1.𝑏
Moreover, these bounds are sharp.
Proposition 3.8. Let 𝑓 (𝑥) ∈ R[𝑥] be separable of degree 𝑛 ⩾ 3 with leading coefficient 𝑏.
Suppose further that all 𝑛 of the roots of 𝑓 ar(e real. T)hen
8 2 ( ) 1/(𝑛−1)( . 𝑀 𝑓sep 𝑓 ) ⩽ · .
𝑛 |𝑏 |
Proposition 3.9. Let 𝑓 (𝑥) ∈ R[𝑥] have degree( 3 and)leading coefficient 𝑏. If 𝑓 (𝑥) hasexactly one real root, then √ 1/2
( ) 𝑀 ( 𝑓 )sep 𝑓 < 3 |𝑏 | .
Proposition 3.10. Suppose that 𝑓 (𝑥) ∈ R[𝑥] has degree 4, leading coefficient 𝑏 and no
real roots. Then
√ ( )1/4
( ) 𝑀 ( 𝑓 )sep 𝑓 ⩽ 2 |𝑏 | .
Moreover, this bound is sharp.
These propositions cover each of the cases stated in Theorem 1.34, so once we have
given proofs of these four propositions, we will have proved Theorem 1.34.
85
Proof of Proposition 3.7. Suppose that the two roots of 𝑓 (𝑥) are 𝛼1 and 𝛼2.
Case 1: First, suppose that 𝑓 (𝑥) has no real roots. In this case, |𝛼( 1 | = |𝛼)2 | and hence,
( ) 1/2𝑀 𝑓
sep( 𝑓 ) = |𝛼 1/2 1/21 − 𝛼2 | ⩽ 2|𝛼1 | = 2|𝛼1 | |𝛼2 | ⩽ 2 | | .𝑏
This bound is achieved by the family of polynomials 𝑃𝑟 (𝑥) = (𝑥 − 𝑖𝑟) (𝑥 + 𝑖𝑟) for 𝑟 ⩾ 1.
Case 2: Second, suppose that 𝑓 (𝑥) has two real roots. Without loss of generality, we can
assume that 𝛼2 ⩾ max(1, |𝛼1 |). If 𝛼1 ⩾ −1, then we have
sep( 𝑓 ) ⩽ 𝛼2 +
𝑀 ( 𝑓 )
1 ⩽ | + 1𝑏 |
and we are done. Else, 𝛼1 < −1 and we now have
sep( 𝑓 ) = 𝛼2 − 𝛼1
and
𝑀 ( 𝑓 )
| | + 1 = −𝛼1𝛼2 + 1.𝑏
We now observe that the inequality sep( ) ⩽ 𝑀 ( 𝑓 )𝑓 | | + 1 is equivalent to the inequality𝑏
𝛼2 − 1 ⩽ −𝛼1(𝛼2 − 1), which is true by virtue of the fact that 𝛼2 ⩾ 1 and 𝛼1 ⩽ −1. This
bound is achieved by the family of polynomials 𝑄𝑟 (𝑥) = (𝑥 + 1) (𝑥 − 𝑟) for 𝑟 ⩾ 1. □
With the proof of Proposition 3.7, we can now re-consider Figures 3.1 and 3.2 with
the added upper bounds.
In Figure 3.12, we can see that the upper bound 𝑦 = 𝑥 + 1 is attained regularly; it only
doesn’t appear sharp for 𝑀 ( 𝑓 ) > 2 for the following reason. If you choose a polynomial
𝑓 (𝑥) with 𝑀 ( 𝑓 ) > 2 where both roots are chosen from the interval [−2, 2], this forces
both roots to have opposite signs and live outside the interval [−1, 1], which artificially
inflates the Mahler measure relative to the separation.
√
In Figure 3.13, we can see that the upper bound 𝑦 = 2 𝑥 is attained regularly.
We next consider the totally real case for polynomials of degree at least 3.
Proof of Proposition 3.8. We may assume that |𝑏 | = 1; if not, replace 𝑀 ( 𝑓 ) everywhere
in this proof by 𝑀 ( 𝑓 )| | and all statements will still be true.𝑏
First, denote the closest root of 𝑓 (𝑥) to 0 by 𝛼 and let 𝑟 = sep( 𝑓 ). Suppose that there
are 𝑠 roots of 𝑓 (𝑥) which are less than 𝛼 and 𝑡 roots of 𝑓 (𝑥) which are greater than 𝛼.
Define ∏𝑡 ∏𝑠
𝑔(𝑥) = (𝑥 − 𝛼 − 𝑟𝑖) · (𝑥 − 𝛼) · (𝑥 − 𝛼 + 𝑟 𝑗)
𝑖=1 𝑗=1
86
Figure 3.12. Mahler measure against separation for monic quadratic polynomials with two
real roots and the sharp upper bound of 𝑦 = 𝑥 + 1.
Figure 3.13. Mahler measure against separation for monic quadratic polynomials with no
real roots and the sharp upper bound of 𝑦 = 2𝑥1/2.
and observe that we have 𝑀 ( 𝑓 ) ⩾ 𝑀 (𝑔) because the roots of 𝑔 are no further from the
origin than the corresponding roots of 𝑓 . Furthermore, we have sep( 𝑓 ) = 𝑟 = sep(𝑔), so it
suffices to prove the Proposition for 𝑔(𝑥).
Let 𝛽 be the closest roo∏t of 𝑔(𝑥) to the origin and write𝑇 ∏𝑆
𝑔(𝑥) = (𝑥 − 𝛽 − 𝑟𝑖) · (𝑥 − 𝛽) · (𝑥 − 𝛽 + 𝑟 𝑗).
𝑖=1 𝑗=1
Since 𝛽 is the closest root of 𝑔(𝑥) to 0, we note that |𝛽 | ⩽ 𝑟/2. We may also assume
without loss of generality that 𝛽 ⩽ 0 (else, we may apply the same proof to 𝑔(−𝑥)). We
now have ∏𝑇 ∏𝑆
𝑀 (𝑔) ⩾ |𝛽 + 𝑟𝑖 | · |𝛽 − 𝑟 𝑗 |
𝑖=1 𝑗=1
87
∏𝑇 ( 𝑟 ) ∏𝑆 ( )
⩾ 𝑟𝑖 − · 𝑟𝑟 𝑗 +
𝑖=1 ∏ 2( 𝑗=)1𝑇 ∏𝑆 ( 2 )
= 𝑟𝑛−1 · 𝑖 − 1 · 𝑗 + 1 . (3.2)
2
𝑖=1 𝑗=1 2
Getting a handle on the lower bound giv(en)in equation (3.2) will require some use of√
the gamma function. We use the fact that Γ 12 = 𝜋 together with the usual fact that
Γ(𝑥) = (𝑥 − 1)Γ(𝑥 − 1) to note that
∏( ) ( )𝑇 1 Γ 𝑇 + 12
𝑖 − =
2 √
=1 𝜋𝑖
and ∏𝑆 ( ) ( )1 2Γ 𝑆 + 32
𝑗 + = √ .
𝑗=1 2 𝜋
Now applying Gautsch(i’s ine)quality( yield)s∏𝑇 1 Γ 𝑇 + 12− √︁Γ(𝑇 + 1) 𝑇!𝑖 = √ > = √︁ ,
∏𝑖=1𝑆 ( 2 ) ( 𝜋 ) 𝜋(𝑇 + 1) 𝜋(𝑇 + 1)31 2Γ 𝑆 + 2 2Γ(𝑆 + 2) 2(𝑆 + 1)!
𝑗 + = √ > = .
1 2
√︁ √︁
𝑗= 𝜋 𝜋(𝑆 + 2) 𝜋(𝑆 + 2)
Now we have
( ) 𝑛−1 · √︁ 𝑇! · √︁2(𝑆 + 1)!𝑀 𝑔 ⩾ 𝑟 (3.3)
𝜋(𝑇 + 1) 𝜋(𝑆 + 2)
from inequality (3.2) and we aim to estimate the right-hand side of this inequality from
below in terms of 𝑛. We have the restrictions 0 ⩽ 𝑆, 𝑇 ⩽ 𝑛 and 𝑆 + 𝑇 + 1 = 𝑛, so we can
replace 𝑆 + 1 in equation (3.3) by 𝑛 − 𝑇 to find
√︁ 𝑇! · √︁2(𝑆 + 1)! √︁ 𝑇! · √︁ 2(𝑛 − 𝑇)!=
𝜋(𝑇 + 1) 𝜋(𝑆 + 2) 𝜋(𝑇 + 1) 𝜋(𝑛 − 𝑇 + 1)
2𝑛!
= (𝑛)√︁
𝜋 ( ()𝑇√︁+ 1) (𝑛 − 𝑇 + 1)𝑇 2𝑛!⩾ 𝑛
𝜋 ⌊ /2⌋ (𝑛/2 + 1) (𝑛 − 𝑛/2 + 1)𝑛
88
(
Now, using the fact that 𝑛
) √ 2𝑛+1⌊ /2⌋ ⩽ gives𝑛 𝜋(2𝑛+1)
√︁ 𝑇! · √︁2(𝑆 + 1)! ( )√︁ 2𝑛!⩾
( + 1) 𝑛𝜋 𝑇 𝜋(𝑆 + 2) 𝜋 ⌊ /2 (𝑛/2 + 1) (𝑛 − 𝑛/2 + 1)𝑛 √⌋
𝑛! · 2𝑛 + 1
⩾ √
𝜋2𝑛−1(𝑛 + 2)
√ √
2 1𝑛(𝑛/𝑒)𝑛𝑒 12𝑛+1 2𝑛 + 1
>
2𝑛−1(𝑛 + 2)
1
𝑛(𝑛/𝑒)𝑛𝑒 12𝑛+1
⩾
2(𝑛−2()𝑛 + 2) ( )𝑛 𝑛( ) 1 2⩾ 4 𝑒 12𝑛+1 1 −2𝑒 𝑛 + 2𝑛 𝑛
⩾ 2.4
2𝑒
by way of Stirling’s approximation and the fact that 𝑛 ⩾ 3.
Finally, we are able to return to (3.3) to find that that
√
𝑛! · 2𝑛 + 1
𝑀 (𝑔) ⩾ 𝑟𝑛−1 · √ −1( ) (3.4)𝜋2𝑛 (𝑛 + 2)𝑛
⩾ sep(𝑔)𝑛−1 𝑛2.4
2𝑒
which concludes the p(roof. I)n particular, it yields1 ( ) 𝑛
𝑛−1 𝑛−1 3/2
sep( ) 𝑀 (𝑔) 2𝑒 (2𝑒) ( 1 8.2 1𝑔 ⩽ ⩽ √ 𝑛 𝑀 𝑔) 𝑛−1 ⩽ 𝑀 (𝑔) 𝑛−12.4 𝑛 2.4𝑛 𝑛−1 𝑛
under the assumption that 𝑛 ⩾ 3. □
With this proof complete, this gives us a chance to revisit Figures 3.3 and 3.9. We can
replicate the results of the degree 3 case in Figure 3.3 with the upper bound given by the
more precise bound (√ −1 )𝑛
( ) 𝜋2 (𝑛 + 2)
1/(𝑛−1)
sep 𝑓 ⩽ √ 𝑀 ( 𝑓 )1/(𝑛−1) (3.5)
𝑛! · 2𝑛 + 1
given as inequality (3.4) in the proof of Proposition 3.8. However, Figure 3.14 indicates
that the constant is not optimal.
Similarly, we replicate the results of the degree 4 case in Figure 3.9 with the upper
bound coming from (3.5) and again, Figure 3.15 demonstrates that the constant is not
optimal. However, we suspect that for large 𝑛, the inequality becomes sharper.
Next, we consider cubic polynomials which have only 1 real root.
89
Figure 3.14. Mahler measure a√︃gainst separation for monic cubic polynomials with 3 real√ √ √
roots and the upper bound 𝑦 = 10 𝜋/(3 7) ∗ 𝑥.
Figure 3.15. Mahler measure again√︁st s√eparation for monic quartic polynomials with 4 real
roots and the upper bound 𝑦 = log( 2 𝜋/3) + 𝑥/2.
Proof of Proposition 3.9. We may assume that |𝑏 | = 1; if not, replace 𝑀 ( 𝑓 ) everywhere
in this proof by 𝑀 ( 𝑓 )| | and all statements will still be true. Suppose that the real root of 𝑓 (𝑥)𝑏
is 𝛼 and without loss of generality, we may assume that 𝛼 ⩾ 0. Let 𝛽 denote the complex
root of 𝑓 (𝑥) with positive imaginary part.
We claim that we may assume thatℜ[𝛽] ⩽ 0. If not, then set 𝛽′ = −ℜ[𝛽] + 𝑖ℑ[𝛽]
and 𝑔(𝑥) = (𝑥 − 𝛼) (𝑥 − 𝛽′) (𝑥 − 𝛽′). Since sep(𝑔) ⩾ sep( 𝑓 ) and 𝑀 (𝑔) = 𝑀 ( 𝑓 ), proving
the proposition for 𝑔(𝑥) will prove it for 𝑓 (𝑥). Hence, we only need prove the proposition
under the assumption thatℜ[𝛽] ⩽ 0.
We next make a few reductions. √
Let 𝑅 = |𝛽 |. Note that if ℑ[𝛽] ⩽ 3𝑅2 , then
√ √
sep( 𝑓 ) ⩽ |𝛽 − 𝛽 | = 2ℑ[𝛽] ⩽ 3𝑅 ⩽ 3𝑀 ( 𝑓 )1/2
and we are done.
90
√
Hence, for the rest of the proof, assume that ℑ[𝛽] > 3𝑅2 . Note that this automatically
implies that |ℜ[𝛽] | < 𝑅2 . This m{eans that √ }
𝛽 ∈ 𝑆 := 𝑧 ∈ C | ℜ[𝑧] ⩽ 0,ℑ[ ] 3|𝑧 |𝑧 ⩾
2
which√, in the real plane, is the slice of the second quadrant bounded by the lines 𝑥 = 0 and
𝑦 = − 3𝑥.
Our next goal is to reduce to the case where 𝛼, 𝛽, and 𝛽 form an equilateral triangle in
the complex plane. Before we can do this, we claim that if 𝑡 is the unique point in R with
𝑡 > ℜ[𝛽] which√forms an equilateral triangle with 𝛽 and 𝛽, then 𝑡√⩾ 0. It can be easily
checked that 𝑡 = 3ℑ[𝛽] + ℜ[𝛽] and using the fact that ℑ[𝛽] ⩾ 3𝑅2 andℜ[𝛽] ⩾ −
𝑅
2
shows that in fact, 𝑡 ⩾ 𝑅2 > 0.
Suppose then, that 𝛼 ⩾ 𝑡. Then set ℎ(𝑥) = (𝑥 − 𝑡) (𝑥 − 𝛽) (𝑥 − 𝛽). Then
sep(ℎ) = 2ℑ[𝛽] = sep(ℎ) and 𝑀 ( 𝑓 ) ⩾ 𝑀 (ℎ), so it suffices to prove the proposition for
ℎ(𝑥). Hence, we may assume that 𝛼 ⩽ 𝑡.
Suppose that 0 ⩽ 𝛼 ⩽ 𝑡 so that |𝛼 − 𝛽 | = sep( 𝑓 ). Let 𝛽′ be the unique complex
number withℜ[𝛽′] = ℜ[𝛽], which has ℑ[𝛽] > 0, and which forms a√n equilateral triangle
with 𝛼 and 𝛽′. Note that this point is the intersection of the lines 𝛼 = 3ℑ[𝑧] + ℜ[𝑧] and
ℜ[𝑧] = ℜ[𝛽]. Note also that ℑ[𝛽′] ⩽ ℑ[𝛽] because
√ √
3ℑ[𝛽′] + ℜ[𝛽′] = 𝛼 ⩽ 𝑡 = 3ℑ[𝛽] + ℜ[𝛽]
andℜ[𝛽′] = ℜ[𝛽]. Set 𝑗 (𝑥) = (𝑥 − 𝛼) (𝑥 − 𝛽′) (𝑥 − 𝛽′) and note that
sep( 𝑗) = |𝛼 − 𝛽 | = sep( 𝑓 ) and 𝑀 ( 𝑗) ⩽ 𝑀 ( 𝑓 ) sinceℜ[𝛽′] = ℜ[𝛽] and ℑ[𝛽′] ⩽ ℑ[𝛽], so
|𝛽′| ⩽ |𝛽 |. Hence, it suffices to prove the proposition for 𝑗 (𝑥) and we may assume that
𝛼 ⩾ 𝑡. √
For the remainder of the proof then, we h√ave 𝛼 = 𝑡 = 3ℑ[𝛽] + ℜ[𝛽]. So 𝛽 lies on
the line in√the complex plane defined by 𝛼 = 3ℑ[𝑧] + ℜ[𝑧] and above the line
ℑ[𝑧] = − 3ℜ[𝑧]. Letting 𝑥√denote the real par√t of 𝛽 and letting 𝑦 denote the imaginary
part of 𝛽, we the√n have 𝛼 = 3𝑦 + 𝑥 and 𝑦 ⩾ − 3𝑥. The intersection of these two lines is
the point −𝛼 + 𝑖 3𝛼2 2 . Since 𝑥 ⩽ 0, i√t must be the case that 𝛽 lies on the line segment
between the points 𝑖√𝛼 and −𝛼2 + 𝑖
3𝛼
2 . This implies that |𝛽 | ⩽ 𝛼 and moreover, that3
√
3𝛼
𝑦 ⩽ .
2
√
Note also that if 𝑦 < 32 , then
√ √
sep( 𝑓 ) ⩽ 2𝑦 < 3 ⩽ 3𝑀 ( 𝑓 )1/2
91
√
and we are done. So we may assume that 𝑦 ⩾ 3√2 .
Hence, we are left to prove that sep( 𝑓 ) ⩽ 3𝑀 ( 𝑓 )1/2 under the assumption that
√ √
3 3𝛼
⩽ 𝑦 ⩽ .
2 2
We consider the ratio
3𝑀 ( 𝑓 ) 3𝛼 |𝛽 |2
⩾ := 𝑌
sep( 𝑓 )2 4𝑦2
and we aim to show that 𝑌 ⩾ 1. We first wish to express 𝑌 in terms of 𝛼 and 𝑦:
3𝛼 |𝛽 |2
𝑌 =
4𝑦2
3𝛼((𝑥
2 + 𝑦2)
=
4𝑦2
√ )
3𝛼 (𝛼 − 3𝑦)2 + 𝑦2
= 2
3 √
4𝑦
3(𝛼 − 2 3𝛼2𝑦 + 4𝑦2𝛼)
= .
4𝑦2
We claim that under our assumptions, 𝑌 is a nondecreasing function of 𝛼. To see this,
note that
√
𝜕𝑌 9𝛼2 − 12 3𝛼𝑦 + 12𝑦2
=
𝜕𝛼 ( 4𝑦2√ )2
3𝛼 − 2 3𝑦
=
4𝑦2
is never negative. Since 𝛼 ⩾ √2 𝑦, we must have
3
2 √3𝛼 − 2 3𝛼𝑦 + 4𝑦2
𝑌 =
4𝑦2
𝛼
4𝑦2 − 4𝑦2 + 4𝑦2 · 2⩾ 2 √ 𝑦4𝑦 3
⩾ 1
√ √
since 𝑦 ⩾ 32 , which concludes the proof that sep( 𝑓 ) ⩽ 3𝑀 ( 𝑓 )
1/2. □
With this proof complete, we have the chance to revisit Figure 3.4 along with its upper
bound. We recreate this in Figure 3.16.
92
Figure 3.16. Mahler measure again√st separation for monic cubic polynomials with only one
real root and the upper bound 𝑦 = 3𝑥.
Here, we see different behavior than we saw in Figures 3.14 and 3.15. In Figure 3.16,
we see that the constant is optimal, though it appears that the exponent is not optimal.
However, the family of polynomi(als √ ) ( √ )
𝑓𝑡 (𝑥) = (𝑥 − 1) 𝑥 − (1 − 𝑡 3 + 𝑖𝑡) 𝑥 − (1 − 𝑡 3 − 𝑖𝑡)
has sep( 𝑓𝑡) ∼ 𝑀 ( 𝑓 )1/2𝑡 , so the exponent is also optimal. As a result, we see that a relation
of the form sep( 𝑓 ) ⩽ 𝑎 · 𝑀 ( 𝑓 )𝑏 is not subtle enough to capture the relation between
separation and Mahler measure for cubic polynomials with signature (1, 1). This also
explains why the proof of Proposition 3.9 is as complicated as it is.
However, the proof of Proposition 3.10 is more natural and this will show up in the
sharpness of the bound we prove.
Proof of Proposition 3.10. For simplicity, assume |𝑏 | = 1. If |𝑏 | ≠ 1, replace every
instance of 𝑀 ( 𝑓 ) by 𝑀 ( 𝑓 )| and the proof is identical. Suppose that the roots of 𝑓 (𝑥) are𝑏 |
𝛼, ?̄?, 𝛽, 𝛽 where ℑ[𝛼],ℑ[𝛽] > 0 and |𝛼 | = 𝑟 ⩽ |𝛽 | = 𝑅. We first note that
sep( 𝑓 ) = min(2ℑ[𝛼], 2ℑ[𝛽], |𝛼 − 𝛽 |).
We have the following two cases:
Case 1: 2𝑟 ⩽ 𝑅
In this case, we note that ( )1/2 √ √
sep( 𝑅𝑓 ) ⩽ 2ℑ[𝛼] ⩽ 2𝑟 ⩽ 2𝑟1/2 = 2𝑟1/2𝑅1/2 ⩽ 2𝑀 ( 𝑓 )1/4.
2
Case 2: 𝑟 ⩽ 𝑅 < 2𝑟
93
√ √
In this case, we first observe that if ℑ[𝛼] < 2 1/2 1/22 𝑟 𝑅 or if ℑ[𝛽] <
2𝑟1/2𝑅1/22 , then
we are done because
√ √
sep( 𝑓 ) ⩽ min(2ℑ[𝛼], 2ℑ[𝛽]) ⩽ 2𝑟1/2𝑅1/2 ⩽ 2𝑀 ( 𝑓 )1/4.
Hence, { √ }
𝛼 ∈ 𝑧 ∈ 2C : |𝑧 | = 𝑟 and ℑ[𝑧] ⩾ 𝑟1/2𝑅1/2 =: 𝑆
2
and { √ }
𝛽 ∈ 𝑧 ∈ 2C : |𝑧 | = 𝑅 and ℑ[𝑧] ⩾ 𝑟1/2𝑅1/2 =: 𝑇.
2
As a result, we have
sep( 𝑓 ) ⩽ |𝛼 − 𝛽 |
⩽ sup |𝑧1 − 𝑧2 |𝑧1∈𝑆𝑧2∈𝑇= (√︂ √ ) ( √︂ √
)
𝑟𝑅
√︂ 𝑟2 − +√︂𝑖 ·
2
𝑟1/2𝑅1/2 − − 2 − 𝑟𝑅 + · 2𝑅 𝑖 𝑟1/2𝑅1/2 
2 2 2 2 
= 𝑟2 − 𝑟𝑅 + 𝑟𝑅𝑅2 − .
2 2
We claim that √︂ √︂ √
𝑟2 − 𝑟𝑅 + 2 − 𝑟𝑅𝑅 ⩽ 2𝑟1/2𝑅1/2.
2 2
To see this, divide both sides of√︂the inequal√︂ity by 𝑟
1/2𝑅1/2 to obtain the equivalent
inequality
𝑟 − 1 + 𝑅 − 1
√
⩽ 2.
𝑅 2 𝑟 2
Observe that this new inequality only depends on the ratio 𝑥 = 𝑅 , which we have bounded
𝑟
by 1 ⩽ 𝑥 < 2. It is now a simple√︂calculus pro√︂blem to show that
1 − 1 1
√
+ 𝑥 − ⩽ 2
𝑥 2 2
√
for all 1 ⩽ 𝑥 < 2 and the proof that sep( 𝑓 ) ⩽ 2𝑀 ( 𝑓 )1/4 is complete.
To see that the bound is sharp, consider the family of polynomials given in Example
1.30. □
94
Finally, we revisit Figure 3.11 and add on the upper bound from Proposition 3.10. As
we predict√ed before beginning the proof, Figure 3.17 shows that the bound
sep( 𝑓 ) ⩽ 2𝑀 ( 𝑓 )1/4 is attained regularly.
Figure 3.17. Logarithmic Mahler measure against logarithmic√separation for monic quartic
polynomials with no real roots and the upper bound 𝑦 = log( 2) + 𝑥/4
We conclude by noting that these proofs are more or less the limits of the elementary
approach. An elementary proof of Conjecture 1.33 is likely possible for quartic
polynomials with signature (2, 1), but will be tedious and long. A full proof of Conjecture
1.33 is likely to require a more clever approach: perhaps an induction which relates sep( 𝑓 )
to sep( 𝑓 ′) and 𝑀 ( 𝑓 ) to 𝑀 ( 𝑓 ′) or an appeal to a sphere-packing problem that has already
been solved. However, it is encouraging for the truth of the conjecture that it holds in
nontrivial, low-degree cases.
95
APPENDIX A
PARAMETER CHOICES FOR SECTION 2.7.4
𝑛 𝑑0 𝑑 𝑎 𝑏 𝑇 𝑍
6 0 2 0.18 0.29 10 4
7 0.539 2.5 0.2 0.28 7 4
8 0.992 3 0.16 0.41 7 3
9 0.882 3.5 0.17 0.4 6 3
10 1.17 4 0.23 0.41 5 3
11 1.674 4.5 0.14 0.37 5 3
12 2.088 5 0.27 0.41 4 3
13 2.255 5.5 0.2 0.37 4 3
14 2.484 6 0.16 0.35 4 3
15 2.958 6.5 0.13 0.34 4 3
16 3.136 7 0.11 0.32 4 3
17 3.904 7.5 0.32 0.42 3 3
18 4.158 8 0.27 0.39 3 3
19 4.544 8.5 0.23 0.36 3 3
20 4.712 9 0.21 0.35 3 3
21 4.86 9.5 0.19 0.33 3 3
22 5.418 10 0.17 0.32 3 3
23 5.369 10.5 0.16 0.31 3 3
24 5.664 11 0.15 0.31 3 3
25 5.858 11.5 0.14 0.3 3 3
26 6.148 12 0.13 0.29 3 3
27 6.66 12.5 0.12 0.29 3 3
28 7.308 13 0.11 0.28 3 3
29 6.776 13.5 0.11 0.28 3 3
30 7.56 14 0.1 0.27 3 3
31 7.074 14.5 0.1 0.27 3 3
32 8.296 15 0.09 0.27 3 3
33 7.614 15.5 0.09 0.26 3 3
34 7.154 16 0.09 0.26 3 3
35 8.758 16.5 0.08 0.26 3 3
96
𝑛 𝑑0 𝑑 𝑎 𝑏 𝑇 𝑍
36 8.268 17 0.08 0.26 3 3
37 7.728 17.5 0.08 0.25 3 3
38 11.454 18 0.44 0.48 2 3
39 11.799 18.5 0.4 0.45 2 3
40 11.968 19 0.38 0.43 2 3
41 11.946 19.5 0.37 0.42 2 3
42 12.462 20 0.35 0.41 2 3
43 12.797 20.5 0.33 0.39 2 3
44 12.936 21 0.32 0.38 2 3
45 13.266 21.5 0.31 0.37 2 3
46 13.596 22 0.3 0.37 2 3
47 13.926 22.5 0.29 0.36 2 3
48 14.256 23 0.28 0.35 2 3
49 14.144 23.5 0.28 0.35 2 3
50 14.464 24 0.27 0.34 2 3
51 15.015 24.5 0.26 0.33 2 3
52 14.868 25 0.26 0.33 2 3
53 15.183 25.5 0.25 0.32 2 3
54 15.006 26 0.25 0.32 2 3
55 16.064 26.5 0.24 0.32 2 3
56 16.64 27 0.23 0.31 2 3
57 16.443 27.5 0.23 0.31 2 3
58 17.29 28 0.22 0.3 2 3
59 17.073 28.5 0.22 0.3 2 3
60 16.836 29 0.21 0.8 3 2
61 16.579 29.5 0.2 0.8 3 2
62 16.588 30 0.19 0.8 3 2
63 16.878 30.5 0.18 0.8 3 2
64 17.168 31 0.17 0.8 3 2
65 18.06 31.5 0.16 0.8 3 2
66 18.972 32 0.15 0.8 3 2
67 17.416 32.5 0.15 0.8 3 2
68 18.644 33 0.14 0.8 3 2
97
𝑛 𝑑0 𝑑 𝑎 𝑏 𝑇 𝑍
69 17.655 33.5 0.14 0.8 3 2
70 19.234 34 0.13 0.8 3 2
71 19.86 34.5 0.15 0.81 3 2
72 18.144 35 0.15 0.81 3 2
73 19.437 35.5 0.14 0.81 3 2
74 18.338 36 0.14 0.81 3 2
75 20.007 36.5 0.13 0.81 3 2
76 18.868 37 0.13 0.81 3 2
77 20.938 37.5 0.12 0.81 3 2
78 19.764 38 0.12 0.81 3 2
79 22.26 38.5 0.11 0.81 3 2
80 21.056 39 0.11 0.81 3 2
81 19.812 39.5 0.11 0.81 3 2
82 23.16 40 0.1 0.81 3 2
83 21.896 40.5 0.1 0.81 3 2
84 20.988 41 0.1 0.81 3 2
85 20.05 41.5 0.1 0.81 3 2
86 21.924 42 0.11 0.82 3 2
87 20.961 42.5 0.11 0.82 3 2
88 24.128 43 0.1 0.82 3 2
89 22.734 43.5 0.1 0.82 3 2
90 21.726 44 0.1 0.82 3 2
91 25.86 44.5 0.09 0.82 3 2
92 24.852 45 0.09 0.82 3 2
93 23.814 45.5 0.09 0.82 3 2
94 22.746 46 0.09 0.82 3 2
95 22.099 46.5 0.09 0.82 3 2
96 26.904 47 0.08 0.82 3 2
97 25.816 47.5 0.08 0.82 3 2
98 25.164 48 0.08 0.82 3 2
99 24.021 48.5 0.08 0.82 3 2
100 23.324 49 0.08 0.82 3 2
101 22.607 49.5 0.08 0.82 3 2
98
𝑛 𝑑0 𝑑 𝑎 𝑏 𝑇 𝑍
102 28.674 50 0.07 0.82 3 2
103 27.987 50.5 0.08 0.83 3 2
104 27.28 51 0.08 0.83 3 2
105 26.052 51.5 0.08 0.83 3 2
106 25.3 52 0.08 0.83 3 2
107 24.528 52.5 0.08 0.83 3 2
108 30.96 53 0.07 0.83 3 2
109 29.697 53.5 0.07 0.83 3 2
110 28.93 54 0.07 0.83 3 2
111 28.143 54.5 0.07 0.83 3 2
112 27.336 55 0.07 0.83 3 2
113 26.509 55.5 0.07 0.83 3 2
114 25.662 56 0.07 0.83 3 2
115 24.795 56.5 0.07 0.83 3 2
116 32.804 57 0.06 0.83 3 2
117 31.977 57.5 0.06 0.83 3 2
118 31.13 58 0.06 0.83 3 2
119 30.263 58.5 0.06 0.83 3 2
120 29.376 59 0.06 0.83 3 2
121 29.05 59.5 0.06 0.83 3 2
122 28.128 60 0.06 0.83 3 2
123 26.595 60.5 0.07 0.84 3 2
124 35.164 61 0.06 0.84 3 2
125 34.257 61.5 0.06 0.84 3 2
126 33.33 62 0.06 0.84 3 2
127 32.383 62.5 0.06 0.84 3 2
128 31.416 63 0.06 0.84 3 2
129 30.429 63.5 0.06 0.84 3 2
130 30.048 64 0.06 0.84 3 2
131 29.026 64.5 0.06 0.84 3 2
132 28.62 65 0.06 0.84 3 2
133 27.563 65.5 0.06 0.84 3 2
134 27.132 66 0.06 0.84 3 2
99
𝑛 𝑑0 𝑑 𝑎 𝑏 𝑇 𝑍
135 38.409 66.5 0.05 0.84 3 2
136 37.392 67 0.05 0.84 3 2
137 36.355 67.5 0.05 0.84 3 2
138 35.964 68 0.05 0.84 3 2
139 34.892 68.5 0.05 0.84 3 2
140 34.476 69 0.05 0.84 3 2
141 33.369 69.5 0.05 0.84 3 2
142 32.928 70 0.05 0.84 3 2
143 31.786 70.5 0.05 0.84 3 2
144 31.32 71 0.05 0.84 3 2
145 30.844 71.5 0.05 0.84 3 2
146 30.358 72 0.05 0.84 3 2
147 38.394 72.5 0.05 0.85 3 2
148 37.948 73 0.05 0.85 3 2
149 36.771 73.5 0.05 0.85 3 2
150 36.3 74 0.05 0.85 3 2
151 35.088 74.5 0.05 0.85 3 2
152 34.592 75 0.05 0.85 3 2
153 34.086 75.5 0.05 0.85 3 2
154 32.824 76 0.05 0.85 3 2
155 32.293 76.5 0.05 0.85 3 2
156 31.752 77 0.05 0.85 3 2
157 31.201 77.5 0.05 0.85 3 2
158 30.64 78 0.05 0.85 3 2
159 30.069 78.5 0.05 0.85 3 2
160 46.56 79 0.04 0.85 3 2
161 45.298 79.5 0.04 0.85 3 2
162 44.802 80 0.04 0.85 3 2
163 43.505 80.5 0.04 0.85 3 2
164 42.984 81 0.04 0.85 3 2
165 42.453 81.5 0.04 0.85 3 2
166 41.912 82 0.04 0.85 3 2
167 40.55 82.5 0.04 0.85 3 2
100
𝑛 𝑑0 𝑑 𝑎 𝑏 𝑇 𝑍
168 39.984 83 0.04 0.85 3 2
169 39.408 83.5 0.04 0.85 3 2
170 38.822 84 0.04 0.85 3 2
171 38.226 84.5 0.04 0.85 3 2
172 37.62 85 0.04 0.85 3 2
173 37.004 85.5 0.04 0.85 3 2
174 36.378 86 0.04 0.85 3 2
175 35.742 86.5 0.04 0.85 3 2
176 35.096 87 0.04 0.85 3 2
177 44.772 87.5 0.04 0.86 3 2
178 44.166 88 0.04 0.86 3 2
179 43.55 88.5 0.04 0.86 3 2
180 42.924 89 0.04 0.86 3 2
181 42.288 89.5 0.04 0.86 3 2
182 41.642 90 0.04 0.86 3 2
183 40.986 90.5 0.04 0.86 3 2
184 40.32 91 0.04 0.86 3 2
185 39.644 91.5 0.04 0.86 3 2
186 38.958 92 0.04 0.86 3 2
187 38.262 92.5 0.04 0.86 3 2
188 37.556 93 0.04 0.86 3 2
189 36.84 93.5 0.04 0.86 3 2
190 36.114 94 0.04 0.86 3 2
191 36.309 94.5 0.04 0.86 3 2
192 35.568 95 0.04 0.86 3 2
193 34.817 95.5 0.04 0.86 3 2
194 34.056 96 0.04 0.86 3 2
195 34.236 96.5 0.04 0.86 3 2
196 33.46 97 0.04 0.86 3 2
197 32.674 97.5 0.04 0.86 3 2
198 31.878 98 0.04 0.86 3 2
199 57.289 98.5 0.03 0.86 3 2
200 56.608 99 0.03 0.86 3 2
101
𝑛 𝑑0 𝑑 𝑎 𝑏 𝑇 𝑍
201 55.917 99.5 0.03 0.86 3 2
202 54.23 100 0.03 0.86 3 2
203 53.514 100.5 0.03 0.86 3 2
204 52.788 101 0.03 0.86 3 2
205 52.052 101.5 0.03 0.86 3 2
206 52.312 102 0.03 0.86 3 2
207 51.561 102.5 0.03 0.86 3 2
208 50.8 103 0.03 0.86 3 2
209 50.029 103.5 0.03 0.86 3 2
210 49.248 104 0.03 0.86 3 2
211 48.457 104.5 0.03 0.86 3 2
212 47.656 105 0.03 0.86 3 2
213 47.886 105.5 0.03 0.86 3 2
214 47.07 106 0.03 0.86 3 2
215 59.907 106.5 0.03 0.87 3 2
216 59.136 107 0.03 0.87 3 2
217 67.3735 107.5 0.389816 0.881816 2 2
218 67.9042 108 0.399038 0.883038 2 2
219 68.4369 108.5 0.408258 0.884258 2 2
220 69.0792 109 0.408477 0.884477 2 2
221 69.4002 109.5 0.417694 0.885694 2 2
222 70.047 110 0.417909 0.885909 2 2
222 69.504 110 0.404909 0.884909 2 2
223 69.824 110.5 0.405123 0.885123 2 2
224 70.144 111 0.405336 0.885336 2 2
225 70.464 111.5 0.405547 0.885547 2 2
226 71.89 112 0.405757 0.885757 2 2
227 72.215 112.5 0.405966 0.885966 2 2
228 72.54 113 0.406173 0.886173 2 2
229 73.986 113.5 0.406379 0.886379 2 2
230 74.316 114 0.406583 0.886583 2 2
231 74.646 114.5 0.406786 0.886786 2 2
232 74.976 115 0.406988 0.886988 2 2
102
𝑛 𝑑0 𝑑 𝑎 𝑏 𝑇 𝑍
233 76.447 115.5 0.407188 0.887188 2 2
234 76.782 116 0.407387 0.887387 2 2
235 77.117 116.5 0.407585 0.887585 2 2
236 77.452 117 0.407782 0.887782 2 2
237 78.948 117.5 0.407977 0.887977 2 2
238 79.288 118 0.408171 0.888171 2 2
239 79.628 118.5 0.408364 0.888364 2 2
240 79.968 119 0.408556 0.888556 2 2
241 81.489 119.5 0.408747 0.888747 2 2
242 81.834 120 0.408936 0.888936 2 2
243 82.179 120.5 0.409124 0.889124 2 2
244 82.524 121 0.409311 0.889311 2 2
245 84.07 121.5 0.409497 0.889497 2 2
246 84.42 122 0.409682 0.889682 2 2
247 84.77 122.5 0.409865 0.889865 2 2
248 85.12 123 0.410048 0.890048 2 2
249 85.47 123.5 0.410229 0.890229 2 2
250 87.046 124 0.410409 0.890409 2 2
251 87.401 124.5 0.410588 0.890588 2 2
252 87.756 125 0.410766 0.890766 2 2
253 88.111 125.5 0.410943 0.890943 2 2
254 88.466 126 0.411119 0.891119 2 2
255 90.072 126.5 0.411294 0.891294 2 2
256 90.432 127 0.411468 0.891468 2 2
257 90.792 127.5 0.411641 0.891641 2 2
258 91.152 128 0.411813 0.891813 2 2
259 91.512 128.5 0.411984 0.891984 2 2
260 93.148 129 0.412154 0.892154 2 2
261 93.513 129.5 0.412323 0.892323 2 2
262 93.878 130 0.412491 0.892491 2 2
263 94.243 130.5 0.412658 0.892658 2 2
264 94.608 131 0.412824 0.892824 2 2
265 96.274 131.5 0.412989 0.892989 2 2
103
𝑛 𝑑0 𝑑 𝑎 𝑏 𝑇 𝑍
266 96.644 132 0.413153 0.893153 2 2
267 97.014 132.5 0.413316 0.893316 2 2
268 97.384 133 0.413479 0.893479 2 2
269 97.754 133.5 0.41364 0.89364 2 2
270 98.124 134 0.413801 0.893801 2 2
271 99.825 134.5 0.41396 0.89396 2 2
272 100.2 135 0.414119 0.894119 2 2
273 100.575 135.5 0.414277 0.894277 2 2
274 100.95 136 0.414434 0.894434 2 2
275 101.325 136.5 0.41459 0.89459 2 2
276 88.14 137 0.494745 0.904745 2 2
277 88.465 137.5 0.4949 0.9049 2 2
278 88.79 138 0.495053 0.905053 2 2
279 89.115 138.5 0.495206 0.905206 2 2
280 90.816 139 0.495358 0.905358 2 2
281 91.146 139.5 0.495509 0.905509 2 2
282 91.476 140 0.49566 0.90566 2 2
283 91.806 140.5 0.495809 0.905809 2 2
284 92.136 141 0.495958 0.905958 2 2
285 93.867 141.5 0.496106 0.906106 2 2
286 94.202 142 0.496253 0.906253 2 2
287 94.537 142.5 0.4964 0.9064 2 2
288 94.872 143 0.496545 0.906545 2 2
289 95.207 143.5 0.49669 0.90669 2 2
290 96.968 144 0.496834 0.906834 2 2
291 97.308 144.5 0.496978 0.906978 2 2
292 97.648 145 0.49712 0.90712 2 2
293 97.988 145.5 0.497262 0.907262 2 2
294 98.328 146 0.497403 0.907403 2 2
295 98.668 146.5 0.497544 0.907544 2 2
296 100.464 147 0.497684 0.907684 2 2
297 100.809 147.5 0.497823 0.907823 2 2
298 101.154 148 0.497961 0.907961 2 2
104
𝑛 𝑑0 𝑑 𝑎 𝑏 𝑇 𝑍
299 101.499 148.5 0.498099 0.908099 2 2
300 101.844 149 0.498236 0.908236 2 2
301 103.67 149.5 0.498372 0.908372 2 2
302 104.02 150 0.498508 0.908508 2 2
303 104.37 150.5 0.498642 0.908642 2 2
304 104.72 151 0.498777 0.908777 2 2
305 105.07 151.5 0.49891 0.90891 2 2
306 105.42 152 0.499043 0.909043 2 2
307 107.281 152.5 0.499176 0.909176 2 2
308 107.636 153 0.499307 0.909307 2 2
309 107.991 153.5 0.499438 0.909438 2 2
310 108.346 154 0.499569 0.909569 2 2
311 108.701 154.5 0.499698 0.909698 2 2
312 109.056 155 0.499827 0.909827 2 2
313 110.952 155.5 0.499956 0.909956 2 2
314 111.312 156 0.500084 0.910084 2 2
315 111.672 156.5 0.500211 0.910211 2 2
316 112.032 157 0.500338 0.910338 2 2
317 112.392 157.5 0.500464 0.910464 2 2
318 112.752 158 0.500589 0.910589 2 2
319 113.112 158.5 0.500714 0.910714 2 2
320 115.048 159 0.500838 0.910838 2 2
321 115.413 159.5 0.500962 0.910962 2 2
322 115.778 160 0.501085 0.911085 2 2
323 116.143 160.5 0.501208 0.911208 2 2
324 116.508 161 0.50133 0.91133 2 2
325 116.873 161.5 0.501451 0.911451 2 2
326 117.238 162 0.501572 0.911572 2 2
327 119.214 162.5 0.501692 0.911692 2 2
328 119.584 163 0.501812 0.911812 2 2
329 119.954 163.5 0.501931 0.911931 2 2
330 120.324 164 0.50205 0.91205 2 2
331 120.694 164.5 0.502168 0.912168 2 2
105
𝑛 𝑑0 𝑑 𝑎 𝑏 𝑇 𝑍
332 121.064 165 0.502286 0.912286 2 2
333 121.434 165.5 0.502403 0.912403 2 2
334 123.45 166 0.502519 0.912519 2 2
335 123.825 166.5 0.502635 0.912635 2 2
336 124.2 167 0.502751 0.912751 2 2
337 124.575 167.5 0.502866 0.912866 2 2
338 124.95 168 0.50298 0.91298 2 2
339 125.325 168.5 0.503094 0.913094 2 2
340 125.7 169 0.503207 0.913207 2 2
341 126.075 169.5 0.50332 0.91332 2 2
342 128.136 170 0.503433 0.913433 2 2
343 128.516 170.5 0.503545 0.913545 2 2
344 128.896 171 0.503656 0.913656 2 2
345 129.276 171.5 0.503767 0.913767 2 2
346 129.656 172 0.503878 0.913878 2 2
347 130.036 172.5 0.503988 0.913988 2 2
348 130.416 173 0.504097 0.914097 2 2
349 130.796 173.5 0.504206 0.914206 2 2
350 132.902 174 0.504315 0.914315 2 2
351 133.287 174.5 0.504423 0.914423 2 2
352 133.672 175 0.504531 0.914531 2 2
353 134.057 175.5 0.504638 0.914638 2 2
354 134.442 176 0.504745 0.914745 2 2
355 134.827 176.5 0.504851 0.914851 2 2
356 135.212 177 0.504957 0.914957 2 2
357 135.597 177.5 0.505062 0.915062 2 2
358 135.982 178 0.505167 0.915167 2 2
359 138.138 178.5 0.505272 0.915272 2 2
360 138.528 179 0.505376 0.915376 2 2
361 138.918 179.5 0.505479 0.915479 2 2
362 139.308 180 0.505583 0.915583 2 2
363 139.698 180.5 0.505685 0.915685 2 2
364 140.088 181 0.505788 0.915788 2 2
106
𝑛 𝑑0 𝑑 𝑎 𝑏 𝑇 𝑍
365 140.478 181.5 0.50589 0.91589 2 2
366 140.868 182 0.505991 0.915991 2 2
367 141.258 182.5 0.506092 0.916092 2 2
368 141.648 183 0.506193 0.916193 2 2
369 143.859 183.5 0.506293 0.916293 2 2
370 144.254 184 0.506393 0.916393 2 2
371 144.649 184.5 0.506493 0.916493 2 2
372 145.044 185 0.506592 0.916592 2 2
373 145.439 185.5 0.506691 0.916691 2 2
374 145.834 186 0.506789 0.916789 2 2
375 146.229 186.5 0.506887 0.916887 2 2
376 146.624 187 0.506984 0.916984 2 2
377 147.019 187.5 0.507081 0.917081 2 2
378 147.414 188 0.507178 0.917178 2 2
379 147.809 188.5 0.507274 0.917274 2 2
380 150.08 189 0.50737 0.91737 2 2
381 150.48 189.5 0.507466 0.917466 2 2
382 150.88 190 0.507561 0.917561 2 2
383 151.28 190.5 0.507656 0.917656 2 2
384 151.68 191 0.50775 0.91775 2 2
385 152.08 191.5 0.507844 0.917844 2 2
386 152.48 192 0.507938 0.917938 2 2
387 152.88 192.5 0.508032 0.918032 2 2
388 153.28 193 0.508125 0.918125 2 2
389 153.68 193.5 0.508217 0.918217 2 2
390 154.08 194 0.508309 0.918309 2 2
391 156.411 194.5 0.508401 0.918401 2 2
392 156.816 195 0.508493 0.918493 2 2
393 157.221 195.5 0.508584 0.918584 2 2
394 157.626 196 0.508675 0.918675 2 2
395 158.031 196.5 0.508766 0.918766 2 2
396 158.436 197 0.508856 0.918856 2 2
397 158.841 197.5 0.508946 0.918946 2 2
107
𝑛 𝑑0 𝑑 𝑎 𝑏 𝑇 𝑍
398 159.246 198 0.509035 0.919035 2 2
399 159.651 198.5 0.509124 0.919124 2 2
400 160.056 199 0.509213 0.919213 2 2
401 160.461 199.5 0.509302 0.919302 2 2
402 160.866 200 0.50939 0.91939 2 2
403 163.262 200.5 0.509478 0.919478 2 2
404 163.672 201 0.509565 0.919565 2 2
405 164.082 201.5 0.509652 0.919652 2 2
406 164.492 202 0.509739 0.919739 2 2
407 164.902 202.5 0.509826 0.919826 2 2
408 165.312 203 0.509912 0.919912 2 2
409 165.722 203.5 0.509998 0.919998 2 2
410 166.132 204 0.510083 0.920083 2 2
411 166.542 204.5 0.510169 0.920169 2 2
412 166.952 205 0.510254 0.920254 2 2
413 167.362 205.5 0.510338 0.920338 2 2
414 167.772 206 0.510423 0.920423 2 2
415 168.182 206.5 0.510507 0.920507 2 2
416 168.592 207 0.51059 0.92059 2 2
417 171.063 207.5 0.510674 0.920674 2 2
418 171.478 208 0.510757 0.920757 2 2
419 171.893 208.5 0.51084 0.92084 2 2
420 172.308 209 0.510922 0.920922 2 2
421 172.723 209.5 0.511005 0.921005 2 2
422 173.138 210 0.511086 0.921086 2 2
423 173.553 210.5 0.511168 0.921168 2 2
424 173.968 211 0.51125 0.92125 2 2
425 174.383 211.5 0.511331 0.921331 2 2
426 174.798 212 0.511411 0.921411 2 2
427 175.213 212.5 0.511492 0.921492 2 2
428 175.628 213 0.511572 0.921572 2 2
429 176.043 213.5 0.511652 0.921652 2 2
430 176.458 214 0.511732 0.921732 2 2
108
𝑛 𝑑0 𝑑 𝑎 𝑏 𝑇 𝑍
431 179.004 214.5 0.511811 0.921811 2 2
432 179.424 215 0.51189 0.92189 2 2
433 179.844 215.5 0.511969 0.921969 2 2
434 180.264 216 0.512048 0.922048 2 2
435 180.684 216.5 0.512126 0.922126 2 2
436 181.104 217 0.512204 0.922204 2 2
437 181.524 217.5 0.512282 0.922282 2 2
438 181.944 218 0.512359 0.922359 2 2
439 182.364 218.5 0.512436 0.922436 2 2
440 182.784 219 0.512513 0.922513 2 2
441 183.204 219.5 0.51259 0.92259 2 2
442 183.624 220 0.512667 0.922667 2 2
443 184.044 220.5 0.512743 0.922743 2 2
444 184.464 221 0.512819 0.922819 2 2
445 184.884 221.5 0.512894 0.922894 2 2
446 185.304 222 0.51297 0.92297 2 2
447 185.724 222.5 0.513045 0.923045 2 2
448 188.36 223 0.51312 0.92312 2 2
449 188.785 223.5 0.513194 0.923194 2 2
450 189.21 224 0.513269 0.923269 2 2
451 189.635 224.5 0.513343 0.923343 2 2
452 190.06 225 0.513417 0.923417 2 2
453 190.485 225.5 0.513491 0.923491 2 2
454 190.91 226 0.513564 0.923564 2 2
455 191.335 226.5 0.513637 0.923637 2 2
456 191.76 227 0.51371 0.92371 2 2
457 192.185 227.5 0.513783 0.923783 2 2
458 192.61 228 0.513855 0.923855 2 2
459 193.035 228.5 0.513927 0.923927 2 2
460 193.46 229 0.513999 0.923999 2 2
461 193.885 229.5 0.514071 0.924071 2 2
462 194.31 230 0.514143 0.924143 2 2
463 194.735 230.5 0.514214 0.924214 2 2
109
𝑛 𝑑0 𝑑 𝑎 𝑏 𝑇 𝑍
464 195.16 231 0.514285 0.924285 2 2
465 195.585 231.5 0.514356 0.924356 2 2
466 198.316 232 0.514426 0.924426 2 2
467 198.746 232.5 0.514497 0.924497 2 2
468 199.176 233 0.514567 0.924567 2 2
469 199.606 233.5 0.514637 0.924637 2 2
470 200.036 234 0.514707 0.924707 2 2
471 200.466 234.5 0.514776 0.924776 2 2
472 200.896 235 0.514845 0.924845 2 2
473 201.326 235.5 0.514914 0.924914 2 2
474 201.756 236 0.514983 0.924983 2 2
475 202.186 236.5 0.515052 0.925052 2 2
476 202.616 237 0.51512 0.92512 2 2
477 203.046 237.5 0.515188 0.925188 2 2
478 203.476 238 0.515256 0.925256 2 2
479 203.906 238.5 0.515324 0.925324 2 2
480 204.336 239 0.515391 0.925391 2 2
481 204.766 239.5 0.515459 0.925459 2 2
482 205.196 240 0.515526 0.925526 2 2
483 205.626 240.5 0.515593 0.925593 2 2
484 206.056 241 0.515659 0.925659 2 2
485 206.486 241.5 0.515726 0.925726 2 2
486 209.322 242 0.515792 0.925792 2 2
487 209.757 242.5 0.515858 0.925858 2 2
488 210.192 243 0.515924 0.925924 2 2
489 210.627 243.5 0.51599 0.92599 2 2
490 211.062 244 0.516055 0.926055 2 2
491 211.497 244.5 0.516121 0.926121 2 2
492 211.932 245 0.516186 0.926186 2 2
493 212.367 245.5 0.51625 0.92625 2 2
494 212.802 246 0.516315 0.926315 2 2
495 213.237 246.5 0.51638 0.92638 2 2
496 213.672 247 0.516444 0.926444 2 2
110
𝑛 𝑑0 𝑑 𝑎 𝑏 𝑇 𝑍
497 214.107 247.5 0.516508 0.926508 2 2
498 214.542 248 0.516572 0.926572 2 2
499 214.977 248.5 0.516635 0.926635 2 2
500 215.412 249 0.516699 0.926699 2 2
501 215.847 249.5 0.516762 0.926762 2 2
502 216.282 250 0.516825 0.926825 2 2
503 216.717 250.5 0.516888 0.926888 2 2
504 217.152 251 0.516951 0.926951 2 2
505 217.587 251.5 0.517014 0.927014 2 2
506 218.022 252 0.517076 0.927076 2 2
507 218.457 252.5 0.517138 0.927138 2 2
Table A.1. Parameter choices which minimize 𝑇 + 𝑍 .
111
APPENDIX B
PYTHON AND SAGE CODE
B.1 Python Code for Section 2.7.4
Here are the sequence of methods which minimize the value of 𝑇 + 𝑍 for a given
parameter 𝑛. For brevity, the version here does not include the clarifying comments which
are present in the actual Jupyter notebook.
from math import sqrt,floor,log
from collections import deque
import decimal
decimal.getcontext().prec = 10**4
def pZero(n):
if (n <= 8):
return 3.0
else:
return 2.0
def K(d,n):
return 2*sqrt(float((2*n)/((n-1)*(n-2))))*(2.032**(1/n)*(1+sqrt(
float(2/((n-2)* pZero(n)**n)))))**d
def star(n):
return (n-2) * 0.5
def qOne(dZero,d,n):
return pZero(n)**(star(n)-dZero)/K(dZero,n)
def validSmall(dZero,d,n):
return (0<=dZero) and (dZero <= star(n)-1.4) and (dZero <= d) and (1
< d) and (d <= star(n)) and (qOne(dZero,d,n)>max(1,K(d,n)**(1/(d-1)
)))
def validLarge(a,b,n):
return (0<a) and (a < b) and (b < 1 - sqrt(float(2*(n + a**2)/n**2))
)
def L(a,b,n):
return sqrt(float(2*(n+a**2)))/(1-b)
112
def D(a,b,n):
return L(a,b,n)/(n - L(a,b,n))
def A(a,b,n):
return 1/a**2
def chiN(a,b,n):
return D(a,b,n)*(A(a,b,n) + 1)+1
def piN(a,b,n):
return (D(a,b,n)*(4 + A(a,b,n)) + 2)*float(log(2))+(D(a,b,n) + 1)*
float(log(n))/2 + n*A(a,b,n)*D(a,b,n)/2
def E(a,b,n):
return 1/(2*(b**2-a**2))
def Z(dZero,d,a,b,n):
return floor((float(log(E(a,b,n))) + 2*float(log(n)) - float(log(L(a
,b,n) - 2)))/float(log(n-1)))+2
def T(dZero,d,a,b,n):
firstQuantity = float(log((chiN(a,b,n) * n * (d - 1) + 1) / (dZero *
(d-1) + d)))/float(log(d))
secondQuantity = float(log(piN(a,b,n)/float(log(K(d,n)**(-1/(d-1))*
qOne(dZero,d,n))) + d / (dZero * (d - 1) + d)))/float(log(d))
return floor(max(firstQuantity ,secondQuantity))+2
def minTPlusZ(nMin,nMax,prec):
toReturn = []
for n in range(nMin,nMax):
nStar = star(n)
aUpper = (2*n**2 - sqrt(4*n**4 - 4*(n**2 - 2*n)*(n**2 - 2)))
/(2*(n**2 - 2))
a = prec
tempMinA=prec
dZero = 0
tempMinDZero = 0
b = a + prec
tempMinB = a + prec
tempMinT = T(dZero,nStar,a,b,n)
tempMinZ = Z(dZero,nStar,a,b,n)
tempMinSum = tempMinT + tempMinZ
113
while ((a <= aUpper) and (tempMinSum > 4)):
while ((dZero <= nStar - 1.4) and (tempMinSum > 4)):
while ((b < 1-sqrt(2*(n + a**2)/n**2)) and (tempMinSum >
4)):
tempT = T(dZero,nStar,a,b,n)
tempZ = Z(dZero,nStar,a,b,n)
if (tempT + tempZ < tempMinSum):
tempMinA = a
tempMinB = b
tempMinDZero = dZero
tempMinT = tempT
tempMinZ = tempZ
tempMinSum = tempT + tempZ
b += prec
dZero += prec * (nStar - 1.4)
b = a + prec
a += prec
dZero = 0
b = a + prec
assert validSmall(tempMinDZero ,nStar,n), "d0,d,n are invalid"
assert validLarge(tempMinA,tempMinB ,n), "a,b,n are invalid"
assert chiN(tempMinA,tempMinB,n) >= 2, "chiN is too small"
assert piN(tempMinA,tempMinB,n) >= 5*log(2) + 2*log(n), "piN is
too small"
toReturn.append([n,tempMinDZero ,nStar,tempMinA,tempMinB,tempMinT
,tempMinZ])
return toReturn
def minNWithMinTPlusZ(nMax,prec):
n = nMax
nStar = star(n)
aUpper = (2*n**2 - sqrt(4*n**4 - 4*(n**2 - 2*n)*(n**2 - 2)))/(2*(n
**2 - 2))
a = aUpper - prec
b = aUpper
dZero = nStar - 1.4
tempTPlusZ = 4
listOfParams = deque([])
while ((n >= 6) and (tempTPlusZ == 4)):
tempT = T(dZero,nStar,a,b,n)
tempZ = Z(dZero,nStar,a,b,n)
tempTPlusZ = tempT + tempZ
114
while ((a > 0) and (tempTPlusZ > 4)):
while ((b > a) and (tempTPlusZ > 4)):
while ((dZero >= 0) and (tempTPlusZ > 4)):
tempT = T(dZero,nStar,a,b,n)
tempZ = Z(dZero,nStar,a,b,n)
tempTPlusZ = tempT + tempZ
if (tempTPlusZ == 4):
assert validSmall(dZero,nStar,n), "d0,d,n are
invalid"
assert validLarge(a,b,n), "a,b,n are invalid"
assert chiN(a,b,n) >= 2, "chiN is too small"
assert piN(a,b,n) >= 5*log(2) + 2*log(n), "piN
is too small"
listOfParams.appendleft([n,dZero,nStar,a,b,tempT
,tempZ])
dZero -= prec*(nStar - 1.4)
b -= prec
dZero = nStar - 1.4
a -= prec
b = aUpper
n -= 1
aUpper = (2*n**2 - sqrt(4*n**4 - 4*(n**2 - 2*n)*(n**2 - 2)))
/(2*(n**2 - 2))
nStar = star(n)
a = aUpper - prec
b = aUpper
dZero = nStar - 1.4
return listOfParams
B.2 Sage Method For Section 2.7.5
Here is the specific command which takes as input a degree 𝑛 and height 𝐻. It finds
every irreducible trinomial 𝐹 (𝑥, 𝑦) with degree 𝑛 and height 𝐻, solves the Thue equation
|𝐹 (𝑥, 𝑦) | = 1, then stores the trinomials and their solution lists in a .csv file called
degree_n_height_H_thue_equations.csv.
import itertools
import csv
R.<x> = ZZ[]
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def TrinomialThueWriter(degree,height):
filename = "thue_equation_solution_data/degree_{}_height_{}
_thue_equations.csv".format(degree,height)
columnHeads = ["Number of Solutions to |F(x,y)| = 1", "Leading
Coefficient", "Middle Coefficient", "Constant Coefficient", "Middle
Degree", "List of Solutions to |F(x,y)| = 1"]
rows = []
# Note that we only need to check positive leading coefficients
since F(x,y) will have the same solutions as -F(x,y).
# Note also that if the leading coefficient is larger than the
absolute value of the constant coefficient , then we will have
already computed the reciprocal polynomial F(y,x). Hence, we can
skip polynomials where the constant coefficient has absolute value
less than the leading coefficient.
for leadCoeff in range(1,height + 1):
for midCoeff in itertools.chain(range(-height ,0),range(1,height
+1)):
for constantCoeff in itertools.chain(range(-height,-
leadCoeff+1), range(leadCoeff ,height+1)):
if (abs(leadCoeff) == height or abs(midCoeff)== height
or abs(constantCoeff)== height) and (gcd(gcd(leadCoeff ,midCoeff),
constantCoeff)==1):
for midDegree in range(1,degree):
P = leadCoeff * x^degree + midCoeff * x^
midDegree + constantCoeff
if P.is_irreducible():
thueInfo = gp.thueinit(P)
negSolns = gp.thue(thueInfo ,-1)
posSolns = gp.thue(thueInfo ,1)
totalSolns = len(negSolns)+len(posSolns)
rows.append([totalSolns , leadCoeff , midCoeff
, constantCoeff , midDegree , gp.concat(posSolns,negSolns)])
with open(filename ,’w’) as csvfile:
csvwriter = csv.writer(csvfile)
csvwriter.writerow(columnHeads)
csvwriter.writerows(rows)
B.3 Sage Methods for Section 3.3
The following methods, written for Sage, produce a specified number of polynomials
of fixed degree whose roots come from a specified region of the complex plane and may
116
satisfy have other specified properties (e.g. exceed a specified bound on the discriminant
or have a particular signature). These methods allow one to then plot those polynomials’
Mahler measures against their separations.
import itertools
# The below function computes the absolute value of the discriminant
based on the entries of roots (accounting for multiple roots).
def AbsoluteDiscriminantFromRoots(roots):
n = len(roots)
return abs(prod([(roots[i] - roots[j])^2 for i,j in itertools.
product(range(n),range(n)) if i < j]))
# On input $n$ (an even integer) and $R$, the function below generates a
set of $n/2$ points uniformly distributed in the box $|\Im[z]| < R,
|\Re[z]| < R$ and returns the list of those points and their
complex conjugate. If a number of real roots is specified , it
chooses that number of real values in the interval $[-R,R]$ and then
chooses the remaining roots from the same complex box and includes
their complex conjugates. Finally, if a lower bound on the (
absolute) discriminant is specified , the method will ensure that the
absolute discriminant of the set of roots is large enough before
returning the set of roots.
def GenerateComplexRootsInBox(n, R, numRealRoots = 0,
discriminantLowerBound = 0):
assert (n - numRealRoots) % 2 ==0, "invalid signature chosen"
listOfRoots = []
while (len(listOfRoots) < numRealRoots):
listOfRoots.append(RR.random_element(-R,R))
while(len(listOfRoots) < n):
listOfRoots.append(CDF.random_element(-R,R,-R,R))
if AbsoluteDiscriminantFromRoots(listOfRoots) >=
discriminantLowerBound:
return listOfRoots
else:
return GenerateComplexRootsInBox(n,R,numRealRoots ,
discriminantLowerBound)
# Same as the previous method, but now chooses points in the annulus $1/
R < |z| < R$
def GenerateComplexRootsInAnnulus(n, R, numRealRoots = 0,
discriminantLowerBound = 0):
117
assert (n - numRealRoots) % 2 ==0, "invalid signature chosen"
assert R > 1, "invalid annulus chosen"
listOfRoots = []
while (len(listOfRoots) < numRealRoots):
testPoint = RR.random_element(-R,R)
if abs(testPoint) >= 1/R:
listOfRoots.append(testPoint)
while(len(listOfRoots) < n):
testPoint = CDF.random_element(-R,R,-R,R)
tpnorm = testPoint.norm()
if tpnorm <= R^2 and tpnorm >= 1/R^2:
listOfRoots.append(testPoint)
listOfRoots.append(testPoint.conj())
if AbsoluteDiscriminantFromRoots(listOfRoots) >=
discriminantLowerBound:
return listOfRoots
else:
return GenerateComplexRootsInAnnulus(n,R,numRealRoots ,
discriminantLowerBound)
# Same as the previous method, but now chooses points in the ball $|z| <
R$
def GenerateComplexRootsInBall(n, R, numRealRoots = 0,
discriminantLowerBound = 0):
assert (n - numRealRoots) % 2 ==0, "invalid signature chosen"
listOfRoots = []
while (len(listOfRoots) < numRealRoots):
listOfRoots.append(RR.random_element(-R,R))
while(len(listOfRoots) < n):
testPoint = CDF.random_element(-R,R,-R,R)
tpnorm = testPoint.norm()
if tpnorm <= R^2:
listOfRoots.append(testPoint)
listOfRoots.append(testPoint.conj())
if AbsoluteDiscriminantFromRoots(listOfRoots) >=
discriminantLowerBound:
return listOfRoots
else:
return GenerateComplexRootsInBall(n,R,numRealRoots ,
discriminantLowerBound)
118
# The below method takes a list of complex numbers as input and outputs
the minimal distance between distinct elements.
def SeparationFromRoots(listOfRoots):
return min([abs(i - j) for i,j in itertools.product(listOfRoots ,
listOfRoots) if i < j])
# The below method takes a list of complex numbers as input and outputs
the Mahler measure of the monic polynomial with those roots.
def MahlerMeasureFromRoots(listOfRoots):
return prod([abs(listOfRoots[i]) for i in range(len(listOfRoots)) if
abs(listOfRoots[i]) > 1])
# The following method takes as input a number of trials to run the
following experiment. Randomly choose numRoots points from the
specified region (default: ball) satisfying the specified
constraints , then plot the Mahler measure of the polynomial with
those points as roots versus the separation of that same polynomial
def PlotMahlerVSep(numTrials ,numRoots ,radius,numRealRoots = 0,
discriminantLowerBound = 0,region = "ball"):
heightVSsepList = []
if region == "ball":
for j in range(numTrials):
roots = GenerateComplexRootsInBall(numRoots,radius,
numRealRoots ,discriminantLowerBound)
mahlerMeasure = MahlerMeasureFromRoots(roots)
sep = SeparationFromRoots(roots)
heightVSsepList.append((mahlerMeasure ,sep))
return point(heightVSsepList ,axes_labels = ["Mahler measure","
separation"])
if region == "annulus":
for j in range(numTrials):
roots = GenerateComplexRootsInAnnulus(numRoots,radius,
numRealRoots ,discriminantLowerBound)
mahlerMeasure = MahlerMeasureFromRoots(roots)
sep = SeparationFromRoots(roots)
heightVSsepList.append((mahlerMeasure ,sep))
return point(heightVSsepList ,axes_labels = ["Mahler measure","
separation"])
if region == "box":
for j in range(numTrials):
roots = GenerateComplexRootsInBox(numRoots,radius,
numRealRoots ,discriminantLowerBound)
119
mahlerMeasure = MahlerMeasureFromRoots(roots)
sep = SeparationFromRoots(roots)
heightVSsepList.append((mahlerMeasure ,sep))
return point(heightVSsepList ,axes_labels = ["Mahler measure","
separation"])
# Same as the previous method, but plotted on log-log axes
def PlotLogMahlerVLogSep(numTrials ,numRoots ,radius,numRealRoots = 0,
discriminantLowerBound = 0,region = "ball"):
heightVSsepList = []
if region == "ball":
for j in range(numTrials):
roots = GenerateComplexRootsInBall(numRoots,radius,
numRealRoots ,discriminantLowerBound)
mahlerMeasure = MahlerMeasureFromRoots(roots)
sep = SeparationFromRoots(roots)
heightVSsepList.append((log(mahlerMeasure),log(sep)))
return point(heightVSsepList ,axes_labels = ["log Mahler measure"
,"log separation"])
if region == "annulus":
for j in range(numTrials):
roots = GenerateComplexRootsInAnnulus(numRoots,radius,
numRealRoots ,discriminantLowerBound)
mahlerMeasure = MahlerMeasureFromRoots(roots)
sep = SeparationFromRoots(roots)
heightVSsepList.append((log(mahlerMeasure),log(sep)))
return point(heightVSsepList ,axes_labels = ["log Mahler measure"
,"log separation"])
if region == "box":
for j in range(numTrials):
roots = GenerateComplexRootsInBox(numRoots,radius,
numRealRoots ,discriminantLowerBound)
mahlerMeasure = MahlerMeasureFromRoots(roots)
sep = SeparationFromRoots(roots)
heightVSsepList.append((log(mahlerMeasure),log(sep)))
return point(heightVSsepList ,axes_labels = ["log Mahler measure"
,"log separation"])
120
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