Complements of discriminants of real parabolic function singularities. II

Abstract

We list all connected components of sets of non-discriminant functions near all {\em parabolic} function singularities (which are the second most important family of singularity classes of smooth functions after {\em simple} singularities). Thus, we prove (and improve in one particular case) all the corresponding conjectures from the previous work \cite{para} with the same title. As an application, we enumerate all {\em local Petrovskii lacunas} near arbitrary parabolic singularities of wavefronts of hyperbolic PDEs. We also show that the complements of the discriminant varieties of the versal deformations of $X_9^{\pm}$ and $P_8^1$ singularities have nontrivial one-dimensional homology groups, in contrast to all simple singularities. These results are applications of a general method for investigating and separating non-singular perturbations of real function singularities. An important part of this method is a computer program that formalizes local Picard--Lefschetz theory and surgeries of Morse functions.