Polynomial Root Distribution and Its Impact on Solutions to Thue Equations
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Date
2024-01-09
Authors
Knapp, Greg
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Publisher
University of Oregon
Abstract
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.
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Keywords
Diophantine approximation, gap principle, number theory, root separation, Thue equation