Van der Waerden/Schrijver-Valiant like Conjectures and Stable (aka Hyperbolic) Homogeneous Polynomials: One Theorem for all

Let p be a homogeneous polynomial of degree n in n variables, p(z1, . . . , zn) = p(Z), Z ∈ C. We call such a polynomial p H-Stable if p(z1, . . . , zn) 6= 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) x1 · · · xn then the following inequality holds ∂ ∂x1 . . . ∂xn p(0, . . . , 0) ≥ Cap(p) ∏ 2≤i≤n ( 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 SchrijverValiant 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. the electronic journal of combinatorics 15 (2008), #R66 1 1 The permanent, the mixed discriminant, the Van Der Waerden conjecture(s) and homogeneous polynomials Recall that an n × n matrix A is called doubly stochastic if it is nonnegative entry-wise and its every column and row sum to one. The set of n × n doubly stochastic matrices is denoted by Ωn. Let Λ(k, n) denote the set of n × n matrices with nonnegative integer entries and row and column sums all equal to k. We define the following subset of rational doubly stochastic matrices: Ωk,n = {k−1A : A ∈ Λ(k, n)}. In a 1989 paper [2] R.B. Bapat defined the set Dn of doubly stochastic n-tuples of n × n matrices. An n-tuple A = (A1, . . . , An) belongs to Dn iff Ai 0, i.e. Ai is a positive semi-definite matrix, 1 ≤ i ≤ n; trAi = 1 for 1 ≤ i ≤ n; ∑n i=1 Ai = I, where I, as usual, stands for the identity matrix. Recall that the permanent of a square matrix A is defined by