{
  "id": "1504.05012",
  "version": "v1",
  "published": "2015-04-20T11:07:52.000Z",
  "updated": "2015-04-20T11:07:52.000Z",
  "title": "Polynomials vanishing on Cartesian products: The Elekes-Szabó Theorem revisited",
  "authors": [
    "Orit E. Raz",
    "Micha Sharir",
    "Frank de Zeeuw"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $F\\in\\mathbb{C}[x,y,z]$ be a constant-degree polynomial,and let $A,B,C\\subset\\mathbb C$ be finite sets of size $n$. We show that $F$ vanishes on at most $O(n^{11/6})$ points of the Cartesian product $A\\times B\\times C$, unless $F$ has a special group-related form. This improves a theorem of Elekes and Szab\\'o [Combinatorica, 2012], and generalizes a result of Raz, Sharir, and Solymosi [Amer. J. Math., to appear]. The same statement holds over $\\mathbb{R}$, and a similar statement holds when $A, B, C$ have different sizes (with a more involved bound replacing $O(n^{11/6})$). This result provides a unified tool for improving bounds in various Erd\\H os-type problems in combinatorial geometry, and we discuss several applications of this kind.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-20T11:07:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C10"
    ],
    "keywords": [
      "cartesian product",
      "polynomials vanishing",
      "similar statement holds",
      "finite sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150405012R"
    }
  }
}