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We classify all matroids with at most 8 elements that have the half-planeproperty, and we provide a list of some matroids on 9 elements that have, andthat do not have the half-plane property. Furthermore, we prove that severalclasses of matroids and polynomials that are motivated by the theory ofsemidefinite programming are closed under taking minors and under passing tofaces of the Newton polytope.

Matroids on Eight Elements with the Half-plane Property and Related Concepts