Thue inequalities with few coefficients

TL;DR

This paper investigates solutions to Thue inequalities for sparse binary forms with few nonzero coefficients, providing bounds that depend on the form's discriminant, sparsity, and the inequality parameter.

Contribution

It establishes new upper bounds on the number of solutions for Thue inequalities with forms having few nonzero coefficients, especially when the discriminant is large.

Findings

Solution count bound of  sm^{2/n} for large discriminant

New upper bound independent of m under certain conditions

Bound depends mainly on sparsity s and parameter m

Abstract

Let $F(x, y)$ be a binary form with integer coefficients, degree $n\geq 3$ and irreducible over the rationals. Suppose that only $s + 1$ of the $n + 1$ coefficients of $F$ are nonzero. We show that the Thue inequality $|F(x,y)|\leq m$ has $\ll sm^{2/n}$ solutions provided that the absolute value of the discriminant $D(F)$ of $F$ is large enough. We also give a new upper bound for the number of solutions of $|F(x,y)|\leq m$, with no restriction on the discriminant of $F$ that depends mainly on $s$ and $m$, and slightly on $n$. Our bound becomes independent of $m$ when $m<|D(F)|^{2/(5(n-1))}$, and also independent of $n$ if $|D(F)|$ is large enough.