{
  "id": "1708.05378",
  "version": "v1",
  "published": "2017-08-17T17:49:18.000Z",
  "updated": "2017-08-17T17:49:18.000Z",
  "title": "Geometry of free loci and factorization of noncommutative polynomials",
  "authors": [
    "J. William Helton",
    "Igor Klep",
    "Jurij Volčič"
  ],
  "comment": "30 pages, includes a table of contents",
  "categories": [
    "math.RA",
    "math.AG"
  ],
  "abstract": "The free singularity locus of a noncommutative polynomial f is defined to be the sequence $Z_n(f)=\\{X\\in M_n^g : \\det f(X)=0\\}$ of hypersurfaces. The main theorem of this article shows that f is irreducible if and only if $Z_n(f)$ is eventually irreducible. A key step in the proof is an irreducibility result for linear pencils. Apart from its consequences to factorization in a free algebra, the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-17T17:49:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13J30",
      "15A22",
      "47A56",
      "14P10",
      "16U30",
      "16R30"
    ],
    "keywords": [
      "noncommutative polynomial",
      "free loci",
      "factorization",
      "free singularity locus",
      "linear matrix inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}