Non-classical polynomials and the inverse theorem

TL;DR

This paper investigates the necessity of non-classical polynomials in the inverse theorem for the Gowers $U^k$-norm over finite fields, providing new results especially for the case when $k=p+1$ and beyond.

Contribution

It characterizes precisely when non-classical polynomials are needed in the inverse theorem for the Gowers $U^k$-norm, including a new result for the case $k=p+1$.

Findings

Bounded functions with large $U^k$-norm correlate with classical polynomials for $k \\leq p+1$.

Non-classical polynomials are necessary for $k \\geq p+2$ in the inverse theorem.

The paper provides a complete characterization of when classical polynomials suffice.

Abstract

In this note we characterize when non-classical polynomials are necessary in the inverse theorem for the Gowers $U^k$-norm. We give a brief deduction of the fact that a bounded function on $\mathbb F_p^n$ with large $U^k$-norm must correlate with a classical polynomial when $k\leq p+1$. To the best of our knowledge, this result is new for $k=p+1$ (when $p>2$). We then prove that non-classical polynomials are necessary in the inverse theorem for the Gowers $U^k$-norm over $\mathbb F_p^n$ for all $k\geq p+2$, completely characterizing when classical polynomials suffice.