{
  "id": "1712.10327",
  "version": "v1",
  "published": "2017-12-12T21:42:37.000Z",
  "updated": "2017-12-12T21:42:37.000Z",
  "title": "On the concavity of a sum of elementary symmetric polynomials",
  "authors": [
    "Xavier Lachaume"
  ],
  "comment": "26 pages, 1 appendix",
  "categories": [
    "math.CA"
  ],
  "abstract": "We introduce a new problem on the elementary symmetric polynomials $\\sigma_k$, stemming from the constraint equations of some modified gravity theory. For which coefficients is a linear combination of $\\sigma_k$ $1/p$-concave, with $0 \\leq k \\leq p$? We establish connections between the $1/p$-concavity and the real-rootedness of some polynomials built on the coefficients. We conjecture that if the restriction of the linear combination to the positive diagonal is a real-rooted polynomial, then the linear combination is $1/p$-concave. Using the theory of hyperbolic polynomials, we show that this would be implied by a short algebraic statement: if the polynomials $P$ and $Q$ of degree $n$ are real-rooted, then $\\sum_{k=0}^n P^{(k)}Q^{(n-k)}$ is real-rooted as well. This is not proven yet. We conjecture more generally that the global $1/p$-concavity is equivalent to the $1/p$-concavity on the positive diagonal. We prove all our guessings for $p=2$. The way is open for further developments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-12T21:42:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "elementary symmetric polynomials",
      "linear combination",
      "positive diagonal",
      "short algebraic statement",
      "coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}