Non-classical polynomials and the inverse theorem

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