On the half-plane property and the Tutte group of a matroid

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Abstract

A multivariate polynomial is stable if it is non-vanishing whenever all variables have positive imaginary parts. A matroid has the weak half-plane property (WHPP) if there exists a stable polynomial with support equal to the set of bases of the matroid. If the polynomial can be chosen with all of its non-zero coefficients equal to one then the matroid has the half-plane property (HPP). We describe a systematic method that allows us to reduce the WHPP to the HPP for large families of matroids. This method makes use of the Tutte group of a matroid. We prove that no projective geometry has the WHPP and that a binary matroid has the WHPP if and only if it is regular. We also prove that T8 and R9 fail to have the WHPP. © 2010 Elsevier Inc.

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Brändén, P., & González D’León, R. S. (2010). On the half-plane property and the Tutte group of a matroid. Journal of Combinatorial Theory. Series B, 100(5), 485–492. https://doi.org/10.1016/j.jctb.2010.04.001

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