{
  "id": "1702.08216",
  "version": "v1",
  "published": "2017-02-27T10:09:03.000Z",
  "updated": "2017-02-27T10:09:03.000Z",
  "title": "On higher-order discriminants",
  "authors": [
    "Vladimir Petrov Kostov"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "For the family of polynomials in one variable $P:=x^n+a_1x^{n-1}+\\cdots +a_n$, $n\\geq 4$, we consider its higher-order discriminant sets $\\{ \\tilde{D}_m=0\\}$, where $\\tilde{D}_m:=$Res$(P,P^{(m)})$, $m=2$, $\\ldots$, $n-2$, and their projections in the spaces of the variables $a^k:=(a_1,\\ldots ,a_{k-1},a_{k+1},\\ldots ,a_n)$. Set $P^{(m)}:=\\sum _{j=0}^{n-m}c_ja_jx^{n-m-j}$, $P_{m,k}:=c_kP-x^mP^{(m)}$. We show that Res$(\\tilde{D}_m,\\partial \\tilde{D}_m/\\partial a_k,a_k)= A_{m,k}B_{m,k}C_{m,k}^2$, where $A_{m,k}=a_n^{n-m-k}$, $B_{m,k}=$Res$(P_{m,k},P_{m,k}')$ if $1\\leq k\\leq n-m$ and $A_{m,k}=a_{n-m}^{n-k}$, $B_{m,k}=$Res$(P^{(m)},P^{(m+1)})$ if $n-m+1\\leq k\\leq n$. The equation $C_{m,k}=0$ defines the projection in the space of the variables $a^k$ of the closure of the set of values of $(a_1,\\ldots ,a_n)$ for which $P$ and $P^{(m)}$ have two distinct roots in common. The polynomials $B_{m,k},C_{m,k}\\in \\mathbb{C}[a^k]$ are irreducible. The result is generalized to the case when $P^{(m)}$ is replaced by a polynomial $P_*:=\\sum _{j=0}^{n-m}b_ja_jx^{n-m-j}$, $0\\neq b_i\\neq b_j\\neq 0$ for $i\\neq j$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-27T10:09:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "polynomial",
      "higher-order discriminant sets",
      "projection",
      "distinct roots"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}