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Nearly-polynomial inverse theorem for the U^d norm in degree d+1

Tomer Milo, Guy Moshkovitz

Abstract

We prove a nearly polynomial inverse theorem for the Gowers $U^d$ norm, over finite fields of non-small characteristic, for polynomials of degree $d+1$. The case of degree $d$ was very recently settled by Milićević and Randelović with a fully polynomial bound. We moreover provide a nearly polynomial inverse theorem for homogeneous polynomials of any degree smaller than $2d$.

Nearly-polynomial inverse theorem for the U^d norm in degree d+1

Abstract

We prove a nearly polynomial inverse theorem for the Gowers norm, over finite fields of non-small characteristic, for polynomials of degree . The case of degree was very recently settled by Milićević and Randelović with a fully polynomial bound. We moreover provide a nearly polynomial inverse theorem for homogeneous polynomials of any degree smaller than .
Paper Structure (15 sections, 20 theorems, 68 equations)

This paper contains 15 sections, 20 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Gowers inverse in low degree
  3. Preliminaries and proof outline
  4. Preliminaries
  5. Proof outline
  6. Correlation lemma, the homogeneous and the degree-$d$ cases
  7. A remark on the case of degree-$d$ polynomials
  8. $\mathop{\mathrm{rk}}\nolimits^*$
  9. Properties of $\mathop{\mathrm{rk}}\nolimits^*$
  10. Lower-degree correlation
  11. Discrete derivatives
  12. Discrete derivatives: Proofs
  13. Proof of Theorem \ref{['thm:d+1-PGI-poly']}
  14. Bias of general polynomials
  15. Main proof

Key Result

Theorem 1.1

For any $g \colon \mathbb{F}^n \to \mathbb{C}$ with $\|g\|_{\infty} \le 1$, for some function $\Phi=\Phi_{d,\mathbb{F}}>0$.

Theorems & Definitions (42)

  • Theorem 1.1: Gowers inverse over finite fields GT09BTZ10TZ10TZ12
  • Theorem 1
  • Theorem 2
  • Theorem 2.1: MZ24
  • Lemma 3.1: average correlation lemma
  • proof
  • Theorem 3.2: Warning's second theorem War35
  • Corollary 3.3
  • proof
  • proof : Proof of Theorem \ref{['thm: homogeneous GI 2d']}
  • ...and 32 more