The asymptotic in Waring's problem over function fields via a singular locus in the circle method

TL;DR

This paper advances the understanding of Waring's problem over function fields by employing algebraic geometry techniques to improve asymptotic estimates, surpassing previous integer-based results.

Contribution

It introduces a novel approach using singular locus bounds and exponential sums over finite fields, enhancing asymptotic results in Waring's problem and related conjectures.

Findings

Stronger asymptotic results over function fields than over integers.

Application of Katz's bounds to exponential sums via tangent space calculations.

Extension of methods to Manin's conjecture for Fermat hypersurfaces.

Abstract

We give results on the asymptotic in Waring's problem over function fields that are stronger than the results obtained over the integers using the main conjecture in Vinogradov's mean value theorem. Similar estimates apply to Manin's conjecture for Fermat hypersurfaces over function fields. Following an idea of Pugin, rather than applying analytic methods to estimate the minor arcs, we treat them as complete exponential sums over finite fields and apply results of Katz, which bound the sum in terms of the dimension of a certain singular locus, which we estimate by tangent space calculations.