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

Will Sawin

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.

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

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.

Paper Structure

This paper contains 11 sections, 48 theorems, 328 equations.

Table of Contents

  1. Introduction
  2. Outline of proof
  3. Acknowledgments
  4. Preliminaries
  5. Algebraic geometry
  6. Preliminaries on $\Delta_{k,p}$
  7. The circle method: setup and major arcs
  8. Minor arcs via the singular locus of exponential sums
  9. Manin's conjecture for Fermat hypersurfaces
  10. The singular locus in the arbitrary hypersurface case
  11. Linear programming for the arbitrary hypersurface case

Key Result

Theorem 1.2

Fix a finite field $\mathbb F_q$ of characteristic $p$ and positive integers $k$ and $s$ such that $2\leq k<p$. Let $L_{k,q}= \frac{\log (k+1)}{\log q}$. For all natural numbers $e$ and polynomials $f \in \mathbb F_q[T]$ of degree $\leq ke$, we have poly-asymptotic as soon as and

Theorems & Definitions (97)

  • Theorem 1.2
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 87 more