Sums and differences of four k-th powers

TL;DR

This paper establishes new upper bounds on the number of ways to represent integers as sums of four k-th powers, using advanced methods like the Determinant method and recent results on integral points.

Contribution

It introduces a novel application of the Determinant method to improve bounds on representations by sums of four k-th powers and extends results to general diagonal forms.

Findings

Upper bound of O_N(B^{c/√k}) for representations as sums of four k-th powers

Improved bounds for representations as sums of four k-th powers of non-negative integers

Extension of bounds to general integral diagonal forms

Abstract

We prove an upper bound for the number of representations of a positive integer $N$ as the sum of four $k$-th powers of integers of size at most $B$, using a new version of the Determinant method developed by Heath-Brown, along with recent results by Salberger on the density of integral points on affine surfaces. More generally we consider representations by any integral diagonal form. The upper bound has the form $O_{N}(B^{c/\sqrt{k}})$, whereas earlier versions of the Determinant method would produce an exponent for $B$ of order $k^{-1/3}$ in this case. Furthermore, we prove that the number of representations of a positive integer $N$ as a sum of four $k$-th powers of non-negative integers is at most $O_{\epsilon}(N^{1/k+2/k^{3/2}+\epsilon})$ for $k \geq 3$, improving upon bounds by Wisdom.