Polynomials with the half-plane property and the support theorems

A polynomial P ( x ) in n complex variables is said to have the half-plane property if P ( x ) ≠ 0 whenever all the variables have positive real parts. The generating polynomial for the set of all spanning trees of a graph G is one example. Motivated by the fact that the edge set of each spanning tree of G is a basis of the graphic matroid induced by G, it is shown by Choe et al. (Adv. Appl. Math. 32 (2004) 88–187) that the support of any homogeneous multiaffine polynomial with the half-plane property constitutes the set of all bases of a matroid. In this paper we show, when all the terms of a polynomial with the half-plane property have degrees of same parity, the support constitutes a jump system which is a generalization of matroids. Open problems and a few directions for further research will also be discussed.