Polynomials with the half-plane property and matroid theory

A polynomial f   is said to have the half-plane property if there is an open half-plane H⊂CH⊂C, whose boundary contains the origin, such that f is non-zero whenever all the variables are in H. This paper answers several open questions relating multivariate polynomials with the half-plane property to matroid theory.(1)We prove that the support of a multivariate polynomial with the half-plane property is a jump system. This answers an open question posed by Choe, Oxley, Sokal and Wagner and generalizes their recent result claiming that the same is true whenever the polynomial is also homogeneous.(2)We prove that a multivariate multi-affine polynomial f∈R[z1,…,zn]f∈R[z1,…,zn] has the half-plane property (with respect to the upper half-plane) if and only if∂f∂zi(x)⋅∂f∂zj(x)−∂2f∂zi∂zj(x)⋅f(x)⩾0 for all x∈Rnx∈Rn and 1⩽i,j⩽n1⩽i,j⩽n. This is used to answer two open questions posed by Choe and Wagner regarding strongly Rayleigh matroids.(3)We prove that the Fano matroid is not the support of a polynomial with the half-plane property. This is the first instance of a matroid which does not appear as the support of a polynomial with the half-plane property and answers a question posed by Choe et al. We also discuss further directions and open problems.