Homogeneous multivariate polynomials with the half-plane property

Details Homogeneous multivariate polynomials with the half-plane property Homogeneous multivariate polynomials with the half-plane property

2002-02-05

arxiv.org/abs/math/0202034v2

Adv. Appl. Math. 32, 88-187 (2004)

Mathematics

Combinatorics

LaTeX2e, 111 pages. Submission includes Mathematica programs niceprincipal.m and nicetransversal.m Version 2 corrects a small

Scientific paper

10.1016/S0196-8858(03)00078-2

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.

Affiliated with

Also associated with

No associations

Rating

Homogeneous multivariate polynomials with the half-plane property does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Homogeneous multivariate polynomials with the half-plane property, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Homogeneous multivariate polynomials with the half-plane property will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-705332