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Polynomials with the half-plane property and matroid theory. (English) Zbl 1128.05014
Let $$H$$ be an open half-plane of the complex plane whose boundary contains the origin. A multivariate polynomial is $$H$$-stable when it is non-zero when all the variables are in $$H$$. Polynomials that are $$H$$-stable for some $$H$$ are said to have half-plane property. A support of the polynomial $$f(z) = \sum_{\alpha \in {\mathbb N}^n} a(\alpha) z^{\alpha}$$ is the set of $$\alpha$$s for which $$a(\alpha) \not = 0$$. A jump system is a generalization of the notions of matroid and delta-matroid. The paper shows that the support of polynomials with half-plane property is always a jump system.
A polynomial is multi-affine if it has degree at most one in each variable. The paper gives a necessary and sufficient condition for a multi-affine polynomial with real coefficients to be $$H$$-stable for $$H$$ the upper half-plane $$\{z \in \mathbb C \mid \text{Im}(z)>0\}$$. The condition is given in terms of an inequality involving partial derivatives. Another result of the paper is the existence of a matroid, namely the Fano matroid, which is not a support of a polynomial with half-plane property.

##### MSC:
 05B35 Combinatorial aspects of matroids and geometric lattices
##### Keywords:
half-plane property; Rayleigh matroid; jump system
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##### References:
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