Abstract
AbstractThe notion of the capacity of a polynomial was introduced by Gurvits around 2005, originally to give drastically simplified proofs of the van der Waerden lower bound for permanents of doubly stochastic matrices and Schrijver’s inequality for perfect matchings of regular bipartite graphs. Since this seminal work, the notion of capacity has been utilised to bound various combinatorial quantities and to give polynomial-time algorithms to approximate such quantities (e.g. the number of bases of a matroid). These types of results are often proven by giving bounds on how much a particular differential operator can change the capacity of a given polynomial. In this paper, we unify the theory surrounding such capacity-preserving operators by giving tight capacity preservation bounds for all nondegenerate real stability preservers. We then use this theory to give a new proof of a recent result of Csikvári, which settled Friedland’s lower matching conjecture.
Highlights
Over the past few decades, the theory of real stable polynomials has found various applications, within combinatorics, probability, computer science, and optimisation
The role that polynomials often play in these applications is that of conceptual unification: various natural operations that one may apply to a given type of object can often be represented as natural operations applied to associated polynomials
For the matching polynomial deletion and contraction correspond to certain derivatives, and for the spanning tree polynomial, this idea extends to the minors of a matroid in general
Summary
Over the past few decades, the theory of real stable polynomials has found various applications, within combinatorics, probability, computer science, and optimisation (e.g. see [6, 11] and references therein). Related inequalities have recently have gained importance through the exciting work on a so-called Hodge theory for matroids [3] Results similar to those discussed here can even be extended to basis generating polynomials of matroids in general (not all of which are real stable); see [24] and [2]. The particular line into which this paper falls begins with the work of the first author, who in a series of papers (e.g. see [20]) gave a vast generalisation of the van der Waerden lower bound for permanents of doubly stochastic matrices and the Schrijver lower bound on the number of perfect matchings of regular graphs He showed that a related inequality holds for real stable polynomials in general, and derives each of the referenced results as corollaries. His inequality describes how much the derivative can affect a particular analytic quantity called the capacity of a polynomial, and our main goal in this paper is to extend this bound to a much larger class of linear operators on polynomials
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