The volume of a region defined by polynomial inequalities

Abstract:

Let $P(x)$ be a polynomial on ${R^n}$ with nonnegative coefficients. We develop a simple necessary and sufficient condition that the set $S = \{ x \in {R^n}|{x_i} \geqq 0,P(x) \leqq 1\}$ shall have finite volume. A corresponding result where $P(x)$ is replaced by a collection of polynomials is an easy corollary. Finally, the necessary and sufficient conditions for the special case that P is a product of linear forms is also given.