Abstract
A polynomial p∈R[z1,…,zn] is real stable if it has no roots in the upper-half complex plane. Gurvits's permanent inequality gives a lower bound on the coefficient of the z1z2…zn monomial of a real stable polynomial p with nonnegative coefficients. This fundamental inequality has been used to attack several counting and optimization problems.Here, we study a more general question: Given a stable multilinear polynomial p with nonnegative coefficients and a set of monomials S, we show that if the polynomial obtained by summing up all monomials in S is real stable, then we can lowerbound the sum of coefficients of monomials of p that are in S. We also prove generalizations of this theorem to (real stable) polynomials that are not multilinear. We use our theorem to give a new proof of Schrijver's inequality on the number of perfect matchings of a regular bipartite graph, generalize a recent result of Nikolov and Singh [23], and give deterministic polynomial time approximation algorithms for several counting problems.
Highlights
Suppose we are given a polynomial p ∈ R[z1, . . . , zn] with nonnegative coefficients
Theory of stable polynomials recently had many applications in multiple areas of mathematics and computer science [Gur06, MSS13, AO15a, AO15b]. Gurvits used this theory to give a new proof of the van der Waerden conjecture [Gur06]
Polynomial optimization problems involving real stable polynomials with nonnegative coefficients can often be turned into concave/convex programs
Summary
Suppose we are given a polynomial p ∈ R[z1, . . . , zn] with nonnegative coefficients. Zn] with nonnegative coefficients we analytically and algorithmically study the quantity q(∂z)p(z)|z=0 = p(∂z)q(z)|z=0 =. Matroid Intersection: Let M1([n], I1) and M2([n], I2) be two matroids on elements [n] = {1, 2, . Theory of stable polynomials recently had many applications in multiple areas of mathematics and computer science [Gur, MSS13, AO15a, AO15b]. Gurvits used this theory to give a new proof of the van der Waerden conjecture [Gur06]. Most notably he proved that for any n-homogeneous real stable polynomial p(z1, . In this paper we prove bounds analogous to (2) for the quantity p(z)q(∂z) when p, q are real stable polynomials. We expect to see many more applications in the future
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