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Homogeneous Multivariate Polynomials with the Half-Plane Property

Young-Bin Choe
Organization: University of Waterloo, Waterloo, Ontario, Canada
James G. Oxley
Organization: Louisiana State University, Baton Rouge, Louisiana
Alan D. Sokal
Organization: New York University
David B. Wagner
Journal / Anthology

Year: 2002

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 P_{B(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.

*Mathematics > Algebra > Polynomials
*Mathematics > Calculus and Analysis > Complex Analysis
*Mathematics > Discrete Mathematics > Combinatorics

graph, matroid, jump system, abstract simplicial complex, spanning tree, basis, generating polynomial, reliability polynomial, Brown-Colbourn conjecture, half-plane property, Hurwitz polynomial, positive rational function, Lee-Yang theorem, Heilmann-Lieb theorem, Grace-Walsh-Szegö coincidence theorem, matrix-tree theorem, electrical network, nonnegative matrix, determinant, permanent