{
  "id": "1701.03901",
  "version": "v1",
  "published": "2017-01-14T10:51:56.000Z",
  "updated": "2017-01-14T10:51:56.000Z",
  "title": "Systems of cubic forms in many variables",
  "authors": [
    "Simon L. Rydin Myerson"
  ],
  "comment": "23 pages, submitted",
  "categories": [
    "math.NT"
  ],
  "abstract": "We consider a system of $R$ cubic forms in $n$ variables, with integer coefficients, which define a smooth complete intersection in projective space. Provided $n\\geq 25R$, we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish. In particular we can handle systems of forms in $O(R)$ variables, previous work having required that $n \\gg R^2$. One conjectures that $n \\geq 6R+1$ should be sufficient. We reduce the problem to an upper bound for the number of solutions to a certain auxiliary inequality. To prove this bound we adapt a method of Davenport.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-14T10:51:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D45",
      "11D72",
      "11G35",
      "14G05"
    ],
    "keywords": [
      "cubic forms",
      "smooth complete intersection",
      "integer coefficients",
      "auxiliary inequality",
      "upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}