On the number of solutions of decomposable form inequalities

TL;DR

This paper extends existing estimates for the number of integer solutions of decomposable form inequalities to a broader class of forms, improving upon classical results especially for binary forms.

Contribution

It generalizes Thunder's 2001 estimate to forms of essentially finite type, enhancing the understanding of solution counts for a wider class of inequalities.

Findings

Extended estimates to forms of essentially finite type

Improved bounds for binary forms over Mahler's 1933 results

Broadened applicability of solution counting methods

Abstract

In 2001 Thunder gave an estimate for the number of integer solutions of decomposable form inequalities under the assumption that the forms are of finite type. The purpose of this article is to generalize this result to forms which are of essentially finite type. In the special case of binary forms this gives an improvement of a result of Mahler from 1933.