Computational Algebraic Geometry pp 267-285 | Cite as

# Complexity of Bezout’s Theorem II Volumes and Probabilities

## Abstract

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.

## Keywords

Condition Number Projective Space Unit Sphere Unitary Group Frobenius Norm## Preview

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