Upper bounds on the distribution of the condition number of singular matrices

We exhibit upper bounds for the probability distribution of the generalized condition number of singular complex matrices. To this end, we develop a new technique to study volumes of tubes about projective varieties in the complex projective space. As a main outcome, we show an upper bound estimate for the volume of the intersection of a tube with an equi-dimensional projective algebraic variety. (c) 2005 Academie des sciences. Published by Elsevier SAS. All rights reserved.