{
  "id": "0708.2130",
  "version": "v2",
  "published": "2007-08-16T05:06:18.000Z",
  "updated": "2007-09-16T14:29:17.000Z",
  "title": "On The Solvability of Bilinear Equations in Finite Fields",
  "authors": [
    "Igor E. Shparlinski"
  ],
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "We consider the equation $$ ab + cd = \\lambda, \\qquad a\\in A, b \\in B, c\\in C, d \\in D, $$ over a finite field $F_q$ of $q$ elements, with variables from arbitrary sets $ A, B, C, D \\subseteq F_q$. The question of solvability of such and more general equations has recently been considered by D. Hart and A. Iosevich, who, in particular, proved that if $$ #A #B #C #D \\gg q^3, $$ then above equation has a solution for any $\\lambda \\in F_q^*$. Here we show that using bounds of multiplicative character sums allows us to extend the class of sets which satisfy this property.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-09-16T14:29:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11L40",
      "11T30"
    ],
    "keywords": [
      "finite field",
      "bilinear equations",
      "solvability",
      "multiplicative character sums",
      "general equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0708.2130S"
    }
  }
}