{
  "id": "1802.10173",
  "version": "v1",
  "published": "2018-02-27T21:30:13.000Z",
  "updated": "2018-02-27T21:30:13.000Z",
  "title": "The product of the eigenvalues of a symmetric tensor",
  "authors": [
    "Luca Sodomaco"
  ],
  "comment": "20 pages, 1 figure",
  "categories": [
    "math.AG"
  ],
  "abstract": "We study E-eigenvalues of a symmetric tensor $f$ of degree $d$ on a finite-dimensional Euclidean vector space $V$, and their relation with the E-characteristic polynomial of $f$. We show that the leading coefficient of the E-characteristic polynomial of $f$, when it has maximum degree, is the $(d-2)$-th power (or the $((d-2)/2)$-th power, respectively when $d$ is odd or even) of the $\\widetilde{Q}$-discriminant, where $\\widetilde{Q}$ is the $d$-th Veronese embedding of the isotropic quadric $Q\\subseteq\\mathbb{P}(V)$. This fact, together with a known formula for the constant term of the E-characteristic polynomial of $f$, leads to a closed formula for the product of the E-eigenvalues of $f$, which generalizes the fact that the determinant of a symmetric matrix is equal to the product of its eigenvalues.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-27T21:30:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M20",
      "15A18",
      "15A69",
      "15A72",
      "65H17"
    ],
    "keywords": [
      "symmetric tensor",
      "e-characteristic polynomial",
      "th power",
      "finite-dimensional euclidean vector space",
      "maximum degree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}