Estimates on the distribution of the condition number of singular matrices (Q870485)

Some upper bounds are estimated for the probability distribution of the condition number of singular complex matrices by using some new techniques to study volumes of tubes about algebraic varieties in complex projective spaces. In this respect the following results are proved: the existence of relations between volumes and Intersection Theory in the presence of singularities, an average Bézout Equality for equidimensional varieties, an upper bound for the volume of a tube about a projective variety, and, as a main outcome, an upper bound estimate for the volume of the intersection of a tube with an equidimensional projective algebraic variety.