Analytic and Combinatorial Features of Stable Polynomials

We investigate two main overarching topics in the theory of stable polynomials.1. Differential and Difference Operators. We first study root properties of differen-tial and difference operators. This includes the location of roots, the spacing betweenroots, and the effect of linear operators on the roots. Our results in this directionare mainly for the purpose of theory, with motivation being derived from previousapplications which rely on such results.2. Applications to Graphs. We next study some examples of the connection betweengraphs and polynomials. This includes a study of polynomial capacity and its relationto bipartite matchings, as well as a study of the roots of the independence polynomialand implications of various properties. Our results in this direction are applications ofthe theory of stable polynomials, and demonstrates their use in combinatorics.In the vein of (1), we extend the root bound of [47] to demonstrate the submodular natureof the roots of real-rooted polynomials. This natural extension leads to further questionson generalizations to multivariate polynomials, which are important steps on the path tounderstanding the barrier arguments of Marcus-Spielman-Srivastava. We further settle aconjecture of Brändén-Krasikov-Shapiro on a finite difference convolution which preservesthe spacing of roots of real-rooted polynomials. This convolution generalizes the one studiedby Marcus-Spielman-Srivastava, and the connections between these results yield interestingopen problems which have not yet been fully explored.Towards (2), we combine the Borcea-Brändén characterization of stability preservers withGurvits’ capacity results to produce a theory of capacity preservers. We use this to give a newproof of Csikvári’s lower bound on the matchings of a biregular bipartite graph. In anotherdirection, we give a new proof of the Chudnovsky-Seymour result on the real-rootedness ofthe independence polynomial of a claw-free graph. We also prove root bounds similar to thatof Heilmann-Lieb for a class of independence polynomials, using an interesting multivariateextension of Godsil’s divisibility relations for the matching polynomial