Thue inequalities with few coefficients

Abstract

Let $F(x,y)$ be a binary form with integer coefficients, degree $n\ge 3$ 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 $|F(x,y)|\le m$ has $\ll s{m}^{2/n}$ 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 $|F(x,y)|\le m$, 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.

arXiv admin note: text overlap with arXiv:1906.03705