Contributions to a conjecture of Mueller and Schmidt on Thue inequalities

TL;DR

This paper improves bounds on the number of solutions to Thue inequalities for certain forms, reducing the dependence from s^2 to a smaller function involving s, logs, and exponential terms, under specific coefficient conditions.

Contribution

It refines the upper bounds on solutions to Thue inequalities, replacing s^2 with more precise functions based on form parameters and coefficient conditions.

Findings

Bound s^2 replaced by max(s log^3 s, s e^{ ext{ extPsi}})

Bound s^2 replaced by s log^{3/2} s under symmetry conditions

Improves understanding of solution counts for forms with specific coefficient structures

Abstract

Let $F(X,Y)=\sum\limits_{i=0}^sa_iX^{r_i}Y^{r-r_i}\in\mathbb{Z}[X,Y]$ be a form of degree $r=r_s\geq 3$, irreducible over $\mathbb{Q}$ and having at most $s+1$ non-zero coefficients. Mueller and Schmidt showed that the number of solutions of the Thue inequality \[ |F(X,Y)|\leq h \] is $\ll s^2h^{2/r}(1+\log h^{1/r})$. They $\textit{conjectured}$ that $s^2$ may be replaced by $s$. Let \[ \Psi = \max_{0\leq i\leq s} \max\left( \sum_{w=0}^{i-1}\frac{1}{r_i-r_w},\sum_{w= i+1}^{s}\frac{1}{r_w-r_i}\right). \] Then we show that $s^2$ may be replaced by $\max(s\log^3s, se^{\Psi})$. We also show that if $|a_0|=|a_s|$ and $|a_i|\leq |a_0|$ for $1\leq i\leq s-1$, then $s^2$ may be replaced by $s\log^{3/2}s$. In particular, this is true if $a_i\in\{-1,1\}$.