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Arithmetic harmonic analysis for smooth quartic Weyl sums: three additive equations

Joerg Bruedern, Trevor D. Wooley

Abstract

We establish the non-singular Hasse principle for systems of three diagonal quartic equations in 32 or more variables, subject to a certain rank condition. Our methods employ the arithmetic harmonic analysis of smooth quartic Weyl sums and also a new estimate for their tenth moment.

Arithmetic harmonic analysis for smooth quartic Weyl sums: three additive equations

Abstract

We establish the non-singular Hasse principle for systems of three diagonal quartic equations in 32 or more variables, subject to a certain rank condition. Our methods employ the arithmetic harmonic analysis of smooth quartic Weyl sums and also a new estimate for their tenth moment.

Paper Structure

This paper contains 9 sections, 15 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. The tenth moment of smooth quartic Weyl sums
  3. Large values estimates
  4. Fourier coefficients and their moments
  5. The transition to moments of smooth Weyl sums
  6. Prelude to the circle method
  7. The minor arc treatment
  8. The major arcs analysis
  9. Appendix: an adjuvant lemma

Key Result

Theorem \oldthetheorem

Let $s\geqslant 32$, and suppose that $(a_{ij})\in {\mathbb Z}^{3\times s}$ is propitious. Then provided that the system (1.1) has non-singular real and $p$-adic solutions for each prime number $p$, one has ${\mathcal{N}}(P)\gg P^{s-12}$.

Theorems & Definitions (27)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 17 more