Elsevier

Advances in Applied Mathematics

Volume 32, Issues 1–2, January–February 2004, Pages 88-187
Advances in Applied Mathematics

Homogeneous multivariate polynomials with the half-plane property

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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.

Keywords

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
Matching polynomial
Grace–Walsh–Szegö coincidence theorem
Matrix-tree theorem
Electrical network
Nonnegative matrix
Determinant
Permanent

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