  A polynomial f is said to have the half-plane property if there is an open
half-plane H, whose boundary contains the origin, such that f is non-zero
whenever all the variables are in H. This paper answers several open questions
regarding multivariate polynomials with the half-plane property and matroid
theory.
  * 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.
  * We characterize multivariate multi-affine polynomial with real coefficients
that have the half-plane property (with respect to the upper half-plane) in
terms of inequalities. This is used to answer two open questions posed by Choe
and Wagner regarding strongly Rayleigh matroids.
  * 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.
