Complexity of Bezout’s Theorem II Volumes and Probabilities
Michael Shub, Steve Smale · Birkhäuser Boston eBooks · 1993
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 D 1/2 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. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.