Complexity of Bezout’s Theorem II Volumes and Probabilities

In this paper we study volume estimates in the space of systems of n homegeneous polynomial equations of fixed degrees d_i with respect to a natural Hermitian structure on the space of such systems invariant under the action of the unitary group. We show that the average number of real roots of real systems is D1/2 where D = Π d_i is the Be zout number. We estimate the volume of the subspace of badly conditioned problems and show that volume is bounded by a small degree polynomial in n, N and D times the reciprocal of the condition number to the fourth power. Here N is the dimension of the space of systems.