On the probability distribution of singular varieties of given corank
Journal of Symbolic Computation
Stochastic perturbations and smooth condition numbers
Journal of Complexity
An Arithmetic Poisson Formula for the multi-variate resultant
Journal of Complexity
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We exhibit some new techniques to study volumes of tubes about algebraic varieties in complex projective spaces. We prove the existence of relations between volumes and Intersection Theory in the presence of singularities. In particular, we can exhibit an average Bezout Equality for equidimensional varieties. We also state an upper bound for the volume of a tube about a projective variety. As a main outcome, we prove an upper bound estimate for the volume of the intersection of a tube with an equidimensional projective algebraic variety. We apply these techniques to exhibit upper bounds for the probability distribution of the generalized condition number of singular complex matrices.