Tutte seminar - David Wagner

Stable polynomials, matroids, and sums of squares

Abstract:

A multivariate polynomial p(x₁,...,x_n) is stable provided that it never vanishes when all variables are complex numbers with positive imaginary part. Spanning tree enumerators of graphs and, more generally, basis enumerators of regular matroids are natural examples. Starting from joint work of mine with Choe, Oxley, and Sokal, and developed extensively by Borcea and Brändén, stable polynomials have found numerous applications in combinatorics, probability and statistical mechanics, complex analysis, and matrix theory. In this talk I will concentrate on the characterization of real stable polynomials by quadratic inequalities, the connection with sums of squares, and some applications of this in matroid theory.