Rational points on an intersection of diagonal forms

TL;DR

This paper establishes an asymptotic formula for counting rational points on intersections of diagonal forms with varying degrees, employing advanced analytic number theory techniques including the Hardy-Littlewood method and recent exponential sum estimates.

Contribution

It extends the understanding of rational points on diagonal varieties by providing new asymptotic formulas and sharper results for specific cases using innovative methods.

Findings

Asymptotic formula for rational points on diagonal intersections

Sharper results for particular degree configurations

Application of recent exponential sum estimates

Abstract

We consider intersections of n diagonal forms of degrees k 1 < $\bullet$ $\bullet$ $\bullet$ < kn, and we prove an asymptotic formula for the number of rational points of bounded height on these varieties. The proof uses the Hardy-Littlewood method and recent breakthroughs on the Vinogradov system. We also give a sharper result for one specific value of (k 1 ,. .. , kn), using a technique due to Wooley and an estimate on exponential sums derived from a recent approach in the van der Corput's method.