Thue Inequalities With Few Coefficients

Abstract

Let F(x, y) be a binary form with integer coefficients, degree n ≥ 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)| ≤ m has ≪sm2/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)| ≤ 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.