Abstract:
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 : \begin{itemize} \item {\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 $(1,1,..,1) \in supp(p)$ ? } \item {\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 $(1,1,..,1) \in CO(supp(p))$ ? } \end{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 .