A Note on Thue Inequalities with Few Coefficients

Abstract

Let F(X,Y)=∑i=0saiXriYr−ri∈Z[X,Y] be a form of degree r≥3, irreducible over 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)|≤h

is ≪s2h2/r(1+logh1/r) . They conjectured that s2 may be replaced by s. In this note we show some instances when s2 may be improved.