Talking About Combinatorial Objects Student Seminar: Matroid Representability via Stable Polynomials

Seminar | November 17 | 1-2 p.m. | 748 Evans Hall

 Jonathan Leake, UC Berkeley

 Department of Mathematics

We call a polynomial in $n$ complex variables stable if it is nonvanishing whenever all the variables lie in the open right half-plane. Such polynomials play a key role in a diverse collection of fields, including differential equations, optimization, and combinatorics. In 2004 [1], it was shown that a stable homogeneous multilinear polynomial has support which forms the set of bases of a matroid. (Here, the support of a polynomial $f$ is the set of monomials for which $f$ has a nonzero coefficient.) As it turns out, polynomially-representable matroids form a class which naturally generalizes both graphic matroids and matroids representable over $\mathbb C$. Also, basic operations on matroids such as deletion, contraction, and dual have natural corresponding operations in the polynomial world. In this talk, we will discuss these connections, present counterexamples to obvious next questions, and briefly mention some applications of polynomial-representability.

[1] Y. Choe, J. Oxley, A. Sokal, D. Wagner, Homogeneous multivariate polynomials with the half-plane property. Adv. in Appl. Math. 32 (2004), no. 1-2, 88–187.