{
  "id": "math/0610954",
  "version": "v2",
  "published": "2006-10-31T03:46:02.000Z",
  "updated": "2006-11-01T15:30:19.000Z",
  "title": "A sharper estimate on the Betti numbers of sets defined by quadratic inequalities",
  "authors": [
    "Saugata Basu",
    "Michael Kettner"
  ],
  "comment": "12 pages, 1 figure, corrected typo",
  "journal": "Discrete Comput. Geom. 39:734-746, 2008",
  "categories": [
    "math.AG",
    "math.AT"
  ],
  "abstract": "In this paper we consider the problem of bounding the Betti numbers, $b_i(S)$, of a semi-algebraic set $S \\subset \\R^k$ defined by polynomial inequalities $P_1 \\geq 0,...,P_s \\geq 0$, where $P_i \\in \\R[X_1,...,X_k]$ and $\\deg(P_i) \\leq 2$, for $1 \\leq i \\leq s$. We prove that for $0\\le i\\le k-1$, \\[ b_i(S) \\le{1/2}(\\sum_{j=0}^{min\\{s,k-i\\}}{{s}\\choose j}{{k+1}\\choose {j}}2^{j}). \\] In particular, for $2\\le s\\le \\frac{k}{2}$, we have \\[ b_i(S)\\le {1/2} 3^{s}{{k+1}\\choose {s}} \\leq {1/2} (\\frac{3e(k+1)}{s})^s. \\] This improves the bound of $k^{O(s)}$ proved by Barvinok. This improvement is made possible by a new approach, whereby we first bound the Betti numbers of non-singular complete intersections of complex projective varieties defined by generic quadratic forms, and use this bound to obtain bounds in the real semi-algebraic case.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-11-01T15:30:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14P10",
      "14P25"
    ],
    "keywords": [
      "betti numbers",
      "quadratic inequalities",
      "sharper estimate",
      "non-singular complete intersections"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....10954B"
    }
  }
}