  Let $p(x_1,...,x_n) =\sum_{(r_1,...,r_n) \in I_{n,n}} a_{(r_1,...,r_n)}
\prod_{1 \leq i \leq n} x_{i}^{r_{i}}$ be homogeneous polynomial of degree $n$
in $n$ real variables with integer nonnegative coefficients. The support of
such polynomial $p(x_1,...,x_n)$ is defined as $supp(p) = \{(r_1,...,r_n) \in
I_{n,n} : a_{(r_1,...,r_n)} \neq 0 \}$ . The convex hull $CO(supp(p))$ of
$supp(p)$ is called the Newton polytope of $p$ . We study the following
decision problems, which are far-reaching generalizations of the classical
perfect matching problem : {itemize} {\bf Problem 1 .} Consider a homogeneous
polynomial $p(x_1,...,x_n)$ of degree $n$ in $n$ real variables with
nonnegative integer coefficients given as a black box (oracle) . {\it Is it
true that $(1,1,..,1) \in supp(p)$ ?} {\bf Problem 2 .} Consider a homogeneous
polynomial $p(x_1,...,x_n)$ of degree $n$ in $n$ real variables with
nonnegative integer coefficients given as a black box (oracle) . {\it Is it
true that $(1,1,..,1) \in CO(supp(p))$ ?} {itemize} We prove that for
hyperbolic polynomials these two problems are equivalent and can be solved by
deterministic polynomial-time oracle algorithms . This result is based on a
"hyperbolic" generalization of Rado theorem .
