Complements of discriminants of real parabolic function singularities. II

TL;DR

This paper classifies the connected components of non-discriminant functions near parabolic singularities, proving conjectures, and applies the results to enumerate local Petrovskii lacunas and analyze homology of discriminant complements.

Contribution

It provides a complete classification of non-discriminant function components near parabolic singularities and introduces a computer-assisted method for analyzing real function singularities.

Findings

Confirmed conjectures on non-discriminant function components near parabolic singularities.

Enumerated local Petrovskii lacunas for wavefronts of hyperbolic PDEs.

Discovered nontrivial homology in discriminant complements of certain singularities.

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.