Maximal and typical topology of real polynomial singularities

- We study the structure of polynomial singularities given by semialgebraic conditions on the jet of maps from the sphere to Euclidean space. We prove upper and lower bounds for the homological complexity of these singularities. The upper bound is proved using a semialgebraic version of stratified Morse Theory for jets. For the lower bound, we prove a general result stating that small continuous perturbations of C 1 manifolds can only enrich their topology. In the case of random maps, we provide asymptotic estimates for the expectation of the homological complexity, generalizing classical results of Edelman-Kostlan-Shub- Smale.