On the half-plane property and the Tutte group of a matroid (Q974475)

Let \(E\) be a finite set and let \(z= (z_e)_{e\in E}\) be a vector of variables labeled by the elements of \(E\). A matroid with ground set \(E\) and set of bases \({\mathcal B}\) is said to have the weak half-plane property if there is a weight function \(a:{\mathcal B}\to\mathbb{C}\setminus\{0\}\) such that the polynomial \(\sum_{B\in{\mathcal B}}a(B)z^B\), where \(z^S= \prod_{e\in S}z_e\) is stable. In this paper, it is proved that no finite projective geometry has the weak half-plane property and that a binary matroid has the weak half-property if and only if it is regular.