{
  "id": "1701.02135",
  "version": "v1",
  "published": "2017-01-09T11:11:47.000Z",
  "updated": "2017-01-09T11:11:47.000Z",
  "title": "On the bias of cubic polynomials",
  "authors": [
    "David Kazhdan",
    "Tamar Ziegler"
  ],
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $V$ be a vector space over a finite field $k=\\mathbb{F} _q$ of dimension $n$. For a polynomial $P:V\\to k$ we define the bias of $P$ to be $$b_1(P)=\\frac {|\\sum _{v\\in V}\\psi (P(V))|}{q^n}$$ where $\\psi :k\\to \\mathbb{C} ^\\star$ is a non-trivial additive character. A. Bhowmick and S. Lovett proved that for any $d\\geq 1$ and $c>0$ there exists $r=r(d,c)$ such that any polynomial $P$ of degree $d$ with $b_1(P)\\geq c$ can be written as a sum $P=\\sum _{i=1}^rQ_iR_i$ where $Q_i,R_i:V\\to k$ are non constant polynomials. We show the validity of a modified version of the converse statement for the case $d=3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-09T11:11:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cubic polynomials",
      "non constant polynomials",
      "vector space",
      "finite field",
      "converse statement"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}