








Homogeneous Multivariate Polynomials with the HalfPlane Property






Organization:  University of Waterloo, Waterloo, Ontario, Canada 
Organization:  Louisiana State University, Baton Rouge, Louisiana 
Organization:  New York University 












A polynomial P in n complex variables is said to have the "halfplane property" (or Hurwitz property) if it is nonvanishing whenever all the variables lie in the open right halfplane. 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 runiform set system. We prove that the support (set of nonzero coefficients) of a homogeneous multiaffine polynomial with the halfplane 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 halfplane property? Not all matroids have the halfplane property, but we find large classes that do: all sixthrootofunity matroids, and a subclass of transversal (or cotransversal) matroids that we call "nice". Furthermore, the class of matroids with the halfplane property is closed under minors, duality, direct sums, 2sums, series and parallel connection, fullrank 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 halfplane property: a determinant construction (exploiting "energy" arguments), and a permanent construction (exploiting the HeilmannLieb theorem on matching polynomials). We conclude with a list of open questions.












graph, matroid, jump system, abstract simplicial complex, spanning tree, basis, generating polynomial, reliability polynomial, BrownColbourn conjecture, halfplane property, Hurwitz polynomial, positive rational function, LeeYang theorem, HeilmannLieb theorem, GraceWalshSzegö coincidence theorem, matrixtree theorem, electrical network, nonnegative matrix, determinant, permanent






http://www.arxiv.org/abs/math.CO/0202034







   
 
