Van der Waerden/Schrijver-Valiant like conjectures and stable (aka hyperbolic) homogeneous polynomials: One theorem for all

Abstract

Let p be a homogeneous polynomial of degree n in n variables, p(z 1,... ,Zn) = p(Z), Z ∈ Cn. We call such a polynomial p H-Stable if p(z1, . . .,Zn) ≠ 0 provided the real parts Re(zi) > 0,1 ≤ i ≤ n. This notion from Control Theory is closely related to the notion of Hyperbolicity used intensively in the PDE theory. The main theorem in this paper states that if p(x1, . . . ,xn) is a homogeneous H-Stable polynomial of degree n with nonnegative coefficients; degp(i) is the maximum degree of the variable xi, Ci = min(degp(i),i) and Cap(p) = inf xi>0,1≤i≤n p(x1,. . . ,xn)/x 1. . . xn then the following inequality holds ∂n/∂x1 . . . ∂xnp(0,. . . , 0) ≥ Cap (p) 2≤i≤ (Ci - 1/Ci)Ci - 1. This inequality is a vast (and unifying) generalization of the Van der Waerden conjecture on the permanents of doubly stochastic matrices as well as the Schrijver-Valiant conjecture on the number of perfect matchings in k-regular bipartite graphs. These two famous results correspond to the H-Stable polynomials which are products of linear forms. Our proof is relatively simple and "noncomputational"" ; it uses just very basic properties of complex numbers and the AM/GM inequality.