{
  "id": "2107.07495",
  "version": "v1",
  "published": "2021-07-15T17:48:27.000Z",
  "updated": "2021-07-15T17:48:27.000Z",
  "title": "Non-classical polynomials and the inverse theorem",
  "authors": [
    "Aaron Berger",
    "Ashwin Sah",
    "Mehtaab Sawhney",
    "Jonathan Tidor"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CO"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-15T17:48:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "inverse theorem",
      "non-classical polynomials",
      "brief deduction",
      "classical polynomials suffice",
      "bounded function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}