Mathematics Theses and Dissertations

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https://hdl.handle.net/1794/2044

This collection contains some of the theses and dissertations produced by students in the University of Oregon Mathematics Graduate Program. Paper copies of these and other dissertations and theses are available through the UO Libraries.

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Browsing Mathematics Theses and Dissertations by Author "Akhtari, Shabnam"

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    A Special Family of Binary Forms, Their Invariant Theory, and Related Computations
    (University of Oregon, 2020-12-08) Dethier, Christophe; Akhtari, Shabnam
    In this manuscript we study the family of diagonalizable forms, a special family of integral binary forms. We begin with a summary of definitions and known results relevant to binary forms, diagonalizable forms, Thue equations, and reduction theory. The Thue--Siegel method is applied to derive an upper bound on the number of solutions to Thue's equation $F(x,y) = 1$, where $F$ is a quartic diagonalizable form with negative discriminant. Computation is used in the argument to handle forms whose discriminant is small in absolute value. These results are applied to bound the number of integral points on a certain family of elliptic curves. A proof is given for an alternative classification of diagonalizable forms using the Hessian determinant. Algebraic restrictions are given on the coefficients of a diagonalizable form and divisibility conditions are given on its discriminant. A reduction theory for the family of diagonalizable forms is given. This theory is used to computationally verify that $F(x,y) = 1$, where $F$ is a quintic diagonalizable form with small discriminant, has few solutions.
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    Linear Recurrence Sequences in Diophantine Analysis
    (University of Oregon, 2022-10-04) Bellah, Elisa; Akhtari, Shabnam
    Diophantine analysis is an area of number theory concerned with finding integral solutions to polynomial equations defined over the rationals, or more generally over a number field. In some cases, it is possible to associate a well behaved recurrence sequence to the solution set of a Diophantine equation, which can be useful in generating explicit results. It is known that the solution set to any norm form equation is naturally associated to a family of linear recurrence sequences. As these sequences have been widely studied, Diophantine problems involving norm forms are well-suited to be studied through their associated sequences. In this dissertation, we use this method to study two such problems.
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    Polynomial Root Distribution and Its Impact on Solutions to Thue Equations
    (University of Oregon, 2024-01-09) Knapp, Greg; Akhtari, Shabnam
    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, integral 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.