{
  "id": "1802.04984",
  "version": "v1",
  "published": "2018-02-14T08:20:57.000Z",
  "updated": "2018-02-14T08:20:57.000Z",
  "title": "On ranks of polynomials",
  "authors": [
    "David Kazhdan",
    "Tamar Ziegler"
  ],
  "categories": [
    "math.AG",
    "math.CO"
  ],
  "abstract": "Let $V$ be a vector space over a field $k, P:V\\to k, d\\geq 3$. We show the existence of a function $C(r,d)$ such that $rank (P)\\leq C(r,d)$ for any field $k,char (k)>d$, a finite-dimensional $k$-vector space $V$ and a polynomial $P:V\\to k$ of degree $d$ such that $rank(\\partial P/\\partial t)\\leq r$ for all $t\\in V-0$. Our proof of this theorem is based on the application of results on Gowers norms for finite fields $k$. We don't know a direct proof in the case when $k=\\mathbb C$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-14T08:20:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "polynomial",
      "vector space",
      "gowers norms",
      "finite fields",
      "direct proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}