Complements of discriminants of real parabolic function singularities. II
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917346555199488 |
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| author | Vassiliev, V. A. |
| author_facet | Vassiliev, V. A. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_12738 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Complements of discriminants of real parabolic function singularities. II Vassiliev, V. A. Algebraic Geometry 14Q30, 14B07, 14P25 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. |
| title | Complements of discriminants of real parabolic function singularities. II |
| topic | Algebraic Geometry 14Q30, 14B07, 14P25 |
| url | https://arxiv.org/abs/2512.12738 |