Representation of integers by sparse binary forms

Abstract

We will give new upper bounds for the number of solutions to the inequalities of the shape | F ( x , y ) | ≤ h |F(x,y)| \leq h , where F ( x , y ) F(x,y) is a sparse binary form, with integer coefficients, and h h is a sufficiently small integer in terms of the discriminant of the binary form F F . Our bounds depend on the number of non-vanishing coefficients of F ( x , y ) F(x,y) . When F F is “really sparse”, we establish a sharp upper bound for the number of solutions that is linear in terms of the number of non-vanishing coefficients. This work will provide affirmative answers to a number of conjectures posed by Mueller and Schmidt in [Trans. Amer. Math. Soc. 303 (1987), pp. 241–255], [Acta Math. 160 (1988), pp. 207–247], in special but important cases.