Integral solutions to systems of diagonal equations

TL;DR

This paper develops an asymptotic formula for counting integral solutions to diagonal equations, including solutions with smooth variables, leveraging recent advances in Waring's problem and Vinogradov's mean value theorem.

Contribution

It introduces improved bounds on the number of variables needed for asymptotic formulas, utilizing recent breakthroughs in analytic number theory.

Findings

Derived asymptotic formulas for solutions to diagonal equations

Extended results to solutions with smooth variables

Reduced variable count requirements using recent theoretical progress

Abstract

In this paper, we obtain an asymptotic formula for the number of integral solutions to a system of diagonal equations. We obtain an asymptotic formula for the number of solutions with variables restricted to smooth numbers as well. We improve the required number of variables compared to previous results by incorporating the recent progress on Waring's problem and the resolution of the main conjecture in Vinogradov's mean value theorem.