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Let $p$ be a homogeneous polynomial of degree $n$ in $n$ variables,$p(z_1,...,z_n) = p(Z)$, $Z \in C^{n}$. We call such a polynomial $p$ {\bfH-Stable} if $p(z_1,...,z_n) \neq 0$ provided the real parts $Re(z_i) > 0, 1\leq i \leq n$. This notion from {\it Control Theory} is closely related to thenotion of {\it Hyperbolicity} used intensively in the {\it PDE} theory.  The main theorem in this paper states that if $p(x_1,...,x_n)$ is ahomogeneous {\bf H-Stable} polynomial of degree $n$ with nonnegativecoefficients; $deg_{p}(i)$ is the maximum degree of the variable $x_i$, $C_i =\min(deg_{p}(i),i)$ and $$ Cap(p) = \inf_{x_i > 0, 1 \leq i \leq n}\frac{p(x_1,...,x_n)}{x_1 ... x_n} $$ then the following inequality holds $$\frac{\partial^n}{\partial x_1... \partial x_n} p(0,...,0) \geq Cap(p) \prod_{2\leq i \leq n} (\frac{C_i -1}{C_i})^{C_{i}-1}. $$  This inequality is a vast (and unifying) generalization of the Van derWaerden conjecture on the permanents of doubly stochastic matrices as well asthe Schrijver-Valiant conjecture on the number of perfect matchings in$k$-regular bipartite graphs. These two famous results correspond to the {\bfH-Stable} polynomials which are products of linear forms.  Our proof is relatively simple and ``noncomputational''; it uses just verybasic properties of complex numbers and the AM/GM inequality.

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