Short survey on stable polynomials, orientations and matchings

Abstract

This is a short survey about the theory of stable polynomials and its applications. It gives self-contained proofs of two theorems of Schrijver. One of them asserts that for a d-regular bipartite graph G on 2n vertices, the number of perfect matchings, denoted by pm(G), satisfiesThe other theorem claims that for even d the number of Eulerian orientations of a d-regular graph G on n vertices, denoted by ε(G), satisfiesTo prove these theorems we use the theory of stable polynomials, and give a common generalization of the two theorems.

Capacity of Lorentzian polynomials and distance to binomial distributions

“…There are several proofs for this inequality. One of them reveals a connection with probability theory ( [3]), namely, a theorem of Hoeffding ([7] Theorem 5, [2] Corollary 5.9) on the probability distribution generated by the coefficients of the polynomial easily implies the required statement. The corresponding statement for Lorentzian polynomials was proved by Brändén, Leake and Pak ( [2]).…”