{
  "id": "1701.07196",
  "version": "v1",
  "published": "2017-01-25T07:54:35.000Z",
  "updated": "2017-01-25T07:54:35.000Z",
  "title": "Polynomial equations in function fields",
  "authors": [
    "Pierre-Yves Bienvenu"
  ],
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "The breakthrough paper of Croot, Lev, Pach \\cite{CLP} on progression-free sets in $\\Z_4^n$ introduced a polynomial method that has generated a wealth of applications, such as Ellenberg and Gijswijt's solutions to the cap set problem \\cite{EG}. Using this method, we bound the size of a set of polynomials over $\\F_q$ of degree less than $n$ that is free of solutions to the equation $\\sum_{i=1}^k a_if_i^r=0$, where the coefficients $a_i$ are polynomials that sum to 0 and the number of variables satisfies $k\\geq 2r^2+1$. The bound we obtain is of the form $q^{cn}$ for some constant $c<1$. This is in contrast to the best bounds known for the corresponding problem in the integers, which offer only a logarithmic saving, but work already with as few as $k\\geq r^2+1$ variables.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-25T07:54:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "function fields",
      "polynomial equations",
      "cap set problem",
      "best bounds",
      "polynomial method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}