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, where F (x, y) is a sparse binary form, with integer coefficients, and h is a sufficiently small integer in terms of the discriminant of the binary form F . Our bounds depend on the number of non-vanishing coefficients of F (x, y). When 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 [18] in special but important cases.2000 Mathematics Subject Classification. 11D45.

Thue Inequalities With Few Coefficients

“…Here we establish two new upper bounds for the number of solutions of (1). The bound given in Theorem 1.1 proves Mueller-Schmidt's conjecture for almost all binary forms with given degree.…”

“…The corollary immediately follows from Theorem 1.2 on noticing that log 1) , so when m satisfies (7), we have that 1) , 1) .…”

“…, n) that is not empty, let (x (i) , y (i) ) ∈ X i be the element with the largest value of y. Consider the set of primitive solutions of (13) that are not (x * , y * ) and with 1 ≤ y ≤ Y minus the elements (x (1) , y (1) ), . .…”

“…Under a similar condition on the discriminant, an upper bound for the number of solutions to (1) for small values of m and almost all forms is given in [1], following previous works by [6] and [7]. That bound is linear in s when the forms are 'very' sparse, namely when n ≥ s 2 .…”

“…Let m be a positive integer. Thue studied in [20] the inequalities (1) 1 ≤ |F (x, y)| ≤ m, known as Thue inequalities, showing that they have finitely many solutions in integers x and y. Mahler [9] showed that Thue inequalities have at most c(F )m 2/n solutions, where c(F ) depends only on F . In this bound the dependence on m is best possible if m is large.…”