Non-classical polynomials and the inverse theorem

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