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{\bf Leonid Gurvits}
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{\bf Van der Waerden/Schrijver-Valiant like Conjectures and Stable (aka Hyperbolic) Homogeneous Polynomials: One Theorem for all}
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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 a such polynomial $p$
{\bf H-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 the notion 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 a
homogeneous {\bf H-Stable} polynomial of degree $n$ with nonnegative
coefficients; $deg_{p}(i)$ is the maximum degree of the variable
$x_i$, $C_i = \min(deg_{p}(i),i)$ and $Cap(p) = \inf_{x_i > 0, 1 \leq
i \leq n} {p(x_1,...,x_n)\over x_1 \cdots x_n}$ then the following
inequality holds $${\partial^n\over\partial x_1...\partial x_n}
p(0,...,0) \geq Cap(p) \prod_{2 \leq i \leq n} 
\bigg({C_i-1\over C_i}\bigg)^{C_{i}-1}.$$

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 {\bf 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.


\bye

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