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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.

05B35 Combinatorial aspects of matroids and geometric lattices
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