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

Abstract

Abstract. The methods of Shub and Smale [SS93] are extended to the class of multihomogeneous systems of polynomial equations, yielding Theorem 1, which is a formula expressing the mean (with respect to a particular distribution on the space of coefficient vectors) number of real roots as a multiple of the mean absolute value of the determinant of a random matrix. Theorem 2 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' [Roj96] characterization of the mean number of real roots of an "unmixed" multihomogeneous system. Theorem 3 gives upper and lower bounds for the mean number of roots, where the lower bound is the square root of the generic number of complex roots, as determined by Bernstein's [Ber75] theorem. These bounds are derived by induction from recursive inequalities given in Theorem 4.

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