The expected number of real roots of a multihomogeneous system of polynomial equations

Abstract

The mean number of roots of a multihomogeneous system of polynomial equations (with respect to a natural distribution on the space of coefficient vectors) is shown to be at least as large as the square root of the generic number of complex roots, as determined by Bernshtein's theorem. We first extend the methods of Shub and Smale to the class of multihomogeneous systems of polynomial equations, yielding Theorem A, which is a formula expressing the mean number of real roots as a multiple of the mean absolute value of the determinant of a random matrix. Theorem B derives closed form expressions for the mean in special cases that include: (a) Shub and Smale's result that the expected number of real roots of the general homogeneous system is the square root of the generic number of complex roots given by Bezout's theorem; (b) Rojas' characterization of the mean number of real roots of an "unmixed" multihomogeneous system. Special properties of normal random variables are exploited to derive Theorem C, which gives recursive inequalities satisfied by the mean, and the bounds on the mean number of real roots (we also give an upper bound) are derived by comparing these with recursive equalities satisfied by the numbers given by Bernshtein's theorem.