On the probability distribution of singular varieties of given corank

We exhibit sharp upper bounds for the probability distribution of the distance from a system of multivariate polynomial equations to the strata of all systems having a critical zero of given corank. We also prove sharp upper bounds for the probability distribution of the condition number of singular systems of multivariate polynomial equations. We finally state a new and sharp technique of the Geometry of Numbers. Using this technique we show that rational systems of multivariate polynomial equations are equidistributed with respect to singular systems having a critical zero of given corank.