| Texto completo | |
| Autor(es): |
Bruce, J. W.
;
Fernandes, M. A. C.
;
Tari, F.
Número total de Autores: 3
|
| Tipo de documento: | Artigo Científico |
| Fonte: | RESEARCH IN THE MATHEMATICAL SCIENCES; v. 12, n. 4, p. 20-pg., 2025-10-17. |
| Resumo | |
Given the germ of a smooth plane curve ({f(x,y)=0},0)subset of(K2,0),K=R,C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\{f(x,y)=0\},0)\subset ({\mathbb {K}}<^>2,0), {\mathbb {K}}={\mathbb {R}}, {\mathbb {C}}$$\end{document}, with an isolated singularity, we define two invariants If\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_f$$\end{document} and Vf is an element of N boolean OR{infinity}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_f\in {\mathbb {N}}\cup \{\infty \}$$\end{document}, which count the number of inflections and vertices (suitably interpreted in the complex case) concentrated at the singular point. The first is an affine invariant, while the second is invariant under similarities of R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}<^>2$$\end{document}, and their analogue for C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}<^>2$$\end{document}. When the curve has no smooth components, these invariants are always finite and bounded. We illustrate our results by computing the range of possible values for these invariants for Arnold's K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{K}$$\end{document}-simple singularities. We also establish a relationship between these invariants, the Milnor number of f and the contact of the curve germ with its 'osculating circle'. (AU) | |
| Processo FAPESP: | 22/06325-8 - Deformações geométricas de singularidades |
| Beneficiário: | Marco Antônio do Couto Fernandes |
| Modalidade de apoio: | Bolsas no Brasil - Pós-Doutorado |
| Processo FAPESP: | 19/07316-0 - Teoria de singularidades e aplicações a geometria diferencial, equações diferenciais e visão computacional |
| Beneficiário: | Farid Tari |
| Modalidade de apoio: | Auxílio à Pesquisa - Temático |