Non-classical polynomials and the inverse theorem

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 $k\le p+1$. 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 $k\ge p+2$, completely characterizing when classical polynomials suffice.

11 pages

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