# L-functions in Analytic Number Theory: Greg Knapp

## Event Recap

A recording of this event is available on mathtube.org.

## Topic

Bounds on the Number of Solutions to Thue Equations

## Details

In 1909, Thue proved that when $F(x,y)$ is an irreducible, homogeneous, polynomial with integer coefficients and degree at least 3, the inequality $\left\| F(x,y) \right\| \leq h$ has finitely many integer-pair solutions for any positive $h$. Because of this result, the inequality $\left\| F(x,y) \right\| \leq h$ is known as Thue’s Inequality. Much work has been done to find sharp bounds on the size and number of integer-pair solutions to Thue’s Inequality, with Mueller and Schmidt initiating the modern approach to this problem in the 1980s. In this talk, I will describe different techniques used by Akhtari and Bengoechea; Baker; Mueller and Schmidt; Saradha and Sharma; and Thomas to make progress on this general problem. After that, I will discuss some improvements that can be made to a counting technique used in association with “the gap principle” and how those improvements lead to better bounds on the number of solutions to Thue’s Inequality.

## Additional Information

N.B.: Please note the unusual day.

**Scientific, Seminar**

**April 10, 2024**

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