m at h . C O ] 1 3 M ay 2 00 8 Van der Waerden / Schrijver - Valiant like Conjectures and Stable ( aka Hyperbolic ) Homogeneous Polynomials : One Theorem for

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 .