{
  "id": "1905.07712",
  "version": "v1",
  "published": "2019-05-19T09:12:54.000Z",
  "updated": "2019-05-19T09:12:54.000Z",
  "title": "A few results concerning the Schur stability of the Hadamard powers and the Hadamard products of complex polynomials",
  "authors": [
    "Michał Góra"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "For a complex polynomial \\[ f\\left( s\\right) =s^{n}+a_{n-1}s^{n-1}+\\ldots+a_{1}s+a_{0}% \\] and for a rational number $p$, we consider the Schur stability problem of the $p$-th Hadamard power of $f$ \\[ f^{\\left[ p\\right] }\\left( s\\right) =s^{n}+a_{n-1}^{p}s^{n-1}+\\ldots +a_{1}^{p}s+a_{0}^{p}\\text{.}% \\] We show that there exist two numbers $p^{\\ast}\\geq0\\geq p_{\\ast}$ such that $f^{\\left[ p\\right] }$ is Schur stable for every $p>p^{\\ast}$ and is not Schur stable for $p<p_{\\ast}$ (or vice versa, depending on $f$). Also, we give simple sufficient conditions for the Schur stability of the Hadamard product of two complex polynomials. Numerical examples complete and illustrate the results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-19T09:12:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26C10",
      "30C15"
    ],
    "keywords": [
      "complex polynomial",
      "hadamard product",
      "results concerning",
      "schur stability problem",
      "th hadamard power"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}