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A polynomial P in n complex variables is said to have the "half-plane property" (or Hurwitz property) if it is nonvanishing whenever all the variables lie in the open right half-plane. Such polynomials arise in combinatorics, reliability theory, electrical circuit theory and statistical mechanics. A particularly important case is when the polynomial is homogeneous and multiaffine: then it is the (weighted) generating polynomial of an r-uniform set system. We prove that the support (set of nonzero coefficients) of a homogeneous multiaffine polynomial with the half-plane property is necessarily the set of bases of a matroid. Conversely, we ask: For which matroids M does the basis generating polynomial P_{B(M)} have the half-plane property? Not all matroids have the half-plane property, but we find large classes that do: all sixth-root-of-unity matroids, and a subclass of transversal (or cotransversal) matroids that we call "nice". Furthermore, the class of matroids with the half-plane property is closed under minors, duality, direct sums, 2-sums, series and parallel connection, full-rank matroid union, and some special cases of principal truncation, principal extension, principal cotruncation and principal coextension. Our positive results depend on two distinct (and apparently unrelated) methods for constructing polynomials with the half-plane property: a determinant construction (exploiting "energy" arguments), and a permanent construction (exploiting the Heilmann-Lieb theorem on matching polynomials). We conclude with a list of open questions.Comment: LaTeX2e, 111 pages. Submission includes Mathematica programs niceprincipal.m and nicetransversal.m Version 2 corrects a small error at the beginning of Appendix B, and makes a few small improvements elsewhere. To appear in Advances in Applied Mathematic

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Tutte defined a k-separation of a matroid M to be a partition ðA; BÞ of the ground set of M such that jAj; jBjXk and rðAÞ þ rðBÞ À rðMÞok: If, for all mon; the matroid M has no mseparations, then M is n-connected. Earlier, Whitney showed that ðA; BÞ is a 1-separation of M if and only if A is a union of 2-connected components of M: When M is 2-connected, Cunningham and Edmonds gave a tree decomposition of M that displays all of its 2-separations. When M is 3-connected, this paper describes a tree decomposition of M that displays, up to a certain natural equivalence, all non-trivial 3-separations of M: r

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We show that, for every integer n greater than two, there is a number N such that every 3-connected binary matroid with at least N elements has a minor that is isomorphic to the cycle matroid of K 3, n , its dual, the cycle matroid of the wheel with n spokes, or the vector matroid of the binary matrix (I n | J n &I n ), where J n is the n_n matrix of all ones.

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