A Note on Thue Inequalities with Few Coefficients

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 $r\ge 3$, irreducible over $\mathbb {Q}$, and having at most $s+1$ nonzero coefficients. Mueller and Schmidt showed that the number of solutions of the Thue inequality

$$ |F(X,Y)|\le h $$

is $\ll s^2h^{2/r}(1+\log h^{1/r})$. They conjectured that $s^2$ may be replaced by s. In this note we show some instances when $s^2$ may be improved.

This paper is dedicated to Krishna Alladi on the occasion of his 60th birthday