Analytic and Combinatorial Features of Stable Polynomials
- Author(s): Leake, Jonathan
- Advisor(s): Holtz, Olga
- et al.
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 between
roots, and the effect of linear operators on the roots. Our results in this direction
are mainly for the purpose of theory, with motivation being derived from previous
applications which rely on such results.
2. Applications to Graphs. We next study some examples of the connection between
graphs and polynomials. This includes a study of polynomial capacity and its relation
to bipartite matchings, as well as a study of the roots of the independence polynomial
and implications of various properties. Our results in this direction are applications of
the theory of stable polynomials, and demonstrates their use in combinatorics.
In the vein of (1), we extend the root bound of  to demonstrate the submodular nature
of the roots of real-rooted polynomials. This natural extension leads to further questions
on generalizations to multivariate polynomials, which are important steps on the path to
understanding the barrier arguments of Marcus-Spielman-Srivastava. We further settle a
conjecture of Brändén-Krasikov-Shapiro on a finite difference convolution which preserves
the spacing of roots of real-rooted polynomials. This convolution generalizes the one studied
by Marcus-Spielman-Srivastava, and the connections between these results yield interesting
open problems which have not yet been fully explored.
Towards (2), we combine the Borcea-Brändén characterization of stability preservers with
Gurvits’ capacity results to produce a theory of capacity preservers. We use this to give a new
proof of Csikvári’s lower bound on the matchings of a biregular bipartite graph. In another
direction, we give a new proof of the Chudnovsky-Seymour result on the real-rootedness of
the independence polynomial of a claw-free graph. We also prove root bounds similar to that
of Heilmann-Lieb for a class of independence polynomials, using an interesting multivariate
extension of Godsil’s divisibility relations for the matching polynomial.