{
  "id": "1302.2358",
  "version": "v2",
  "published": "2013-02-10T20:23:27.000Z",
  "updated": "2013-07-07T20:40:55.000Z",
  "title": "A Real Nullstellensatz for Free Modules",
  "authors": [
    "Jaka Cimpric"
  ],
  "comment": "v1 7 pages. v2 9 pages: revised abstract, extended introduction and references. To appear in J. Algebra",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $A$ be the algebra of all $n \\times n$ matrices with entries from $\\RR[x_1,\\ldots,x_d]$ and let $G_1,\\ldots,G_m,F \\in A$. We will show that $F(a)v=0$ for every $a \\in \\RR^d$ and $v \\in \\RR^n$ such that $G_i(a)v=0$ for all $i$ if and only if $F$ belongs to the smallest real left ideal of $A$ which contains $G_1,\\ldots,G_m$. Here a left ideal $J$ of $A$ is real if for every $H_1,\\ldots,H_k \\in A$ such that $H_1^T H_1+\\ldots+H_k^T H_k \\in J+J^T$ we have that $H_1,\\ldots,H_k \\in J$. We call this result the one-sided Real Nullstellensatz for matrix polynomials. We first prove by induction on $n$ that it holds when $G_1,\\ldots,G_m,F$ have zeros everywhere except in the first row. This auxiliary result can be formulated as a Real Nullstellensatz for the free module $\\RR[x_1,\\ldots,x_d]^n$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-07-07T20:40:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13J30"
    ],
    "keywords": [
      "free module",
      "smallest real left ideal",
      "one-sided real nullstellensatz",
      "matrix polynomials",
      "first row"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1302.2358C"
    }
  }
}