Homogeneous multivariate polynomials with the half-plane property

Young Bin Choe, James G. Oxley, Alan D. Sokal, David G. Wagner

    Research output: Contribution to journalArticle

    Abstract

    A polynomial P in n complex variables is said to have the "half-plane property" (or Hurwitz property) if it is nonvanishing whenever all the variables lie in the open right half-plane. Such polynomials arise in combinatorics, reliability theory, electrical circuit theory and statistical mechanics. A particularly important case is when the polynomial is homogeneous and multiaffine: then it is the (weighted) generating polynomial of an r-uniform set system. We prove that the support (set of nonzero coefficients) of a homogeneous multiaffine polynomial with the half-plane property is necessarily the set of bases of a matroid. Conversely, we ask: For which matroids M does the basis generating polynomial PB(M) have the half-plane property? Not all matroids have the half-plane property, but we find large classes that do: all sixth-root-of-unity matroids, and a subclass of transversal (or cotransversal) matroids that we call "nice." Furthermore, the class of matroids with the half-plane property is closed under minors, duality, direct sums, 2-sums, series and parallel connection, full-rank matroid union, and some special cases of principal truncation, principal extension, principal cotruncation and principal coextension. Our positive results depend on two distinct (and apparently unrelated) methods for constructing polynomials with the half-plane property: a determinant construction (exploiting "energy" arguments), and a permanent construction (exploiting the Heilmann-Lieb theorem on matching polynomials). We conclude with a list of open questions.

    Original languageEnglish (US)
    Pages (from-to)88-187
    Number of pages100
    JournalAdvances in Applied Mathematics
    Volume32
    Issue number1-2
    DOIs
    StatePublished - Jan 2004

    Fingerprint

    Multivariate Polynomials
    Homogeneous Polynomials
    Matroid
    Half-plane
    Polynomials
    Polynomial
    Matching Polynomial
    Reliability Theory
    Electrical Circuits
    Set Systems
    Reliability theory
    Complex Variables
    Roots of Unity
    Statistical mechanics
    Combinatorics
    Direct Sum
    Statistical Mechanics
    Circuit theory
    Truncation
    Minor

    ASJC Scopus subject areas

    • Applied Mathematics
    • Discrete Mathematics and Combinatorics

    Cite this

    Homogeneous multivariate polynomials with the half-plane property. / Choe, Young Bin; Oxley, James G.; Sokal, Alan D.; Wagner, David G.

    In: Advances in Applied Mathematics, Vol. 32, No. 1-2, 01.2004, p. 88-187.

    Research output: Contribution to journalArticle

    Choe, Young Bin ; Oxley, James G. ; Sokal, Alan D. ; Wagner, David G. / Homogeneous multivariate polynomials with the half-plane property. In: Advances in Applied Mathematics. 2004 ; Vol. 32, No. 1-2. pp. 88-187.
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