Counting integer points on affine surfaces with a side condition

TL;DR

This paper extends the determinant method to uniformly bound integral points on affine surfaces with polynomial congruence conditions, impacting problems like points on diagonal quadrics and sums of unlike powers.

Contribution

It introduces a new uniform upper bound for integral points on affine surfaces under side conditions, generalizing previous work and applying it to specific Diophantine problems.

Findings

Established a uniform bound for integral points with side conditions

Applied the bound to diagonal quadric surfaces

Provided insights into representing integers as sums of four unlike powers

Abstract

We extend work of Heath-Brown and Salberger, based on the determinant method, to provide a uniform upper bound for the number of integral points of bounded height on an affine surface, which are subject to a polynomial congruence condition. This is applied to get a new uniform bound for points on diagonal quadric surfaces, and to a problem about the representation of integers as a sum of four unlike powers.