Thèse de Doctorat
DOI
https://doi.org/10.11606/T.55.2019.tde-21112019-154256
Document
Auteur
Nom complet
Euripides Alves da Silva
Unité de l'USP
Domain de Connaissance
Date de Soutenance
Editeur
São Carlos, 1982
Directeur
Jury
Favaro, Luiz Antonio (Président)
Carneiro, Mario Jorge Dias
Qualifik, Paul
Teixeira, Marco Antonio
Titre en portugais
CLASSIFICAÇÃO DE PARES BI-ESTÁVEIS POR R-ÁLGEBRAS
Mots-clés en portugais
Não disponível
Resumé en portugais
Não disponível
Titre en anglais
Not available
Mots-clés en anglais
Not available
Resumé en anglais
Let f:Rn, 0 → Rp a C map-germ and let us consider the local algebra of order k, QΓ (f) = En / f * Mp + Mk+1n associated with germ f, where En is the ring of germs g : Rn , 0 → R and Mn, is the maximal ideal of germs g : Rn, 0 → R, 0. The Classification ot Stable Germs Theorem through the local algebras is classic: "If f and g are stable, them f and g are A-equivalent if, and only if, the associated algebras are isomorphic"; see, J. Mather [10]. In [3], J.P. Dufour has introduced the notion of stabliitv for couples of germs (f1, f2) : Rn, 0 → Rp x Rq, 0 and has studied the problem of deformations and classification in particular cases, with his own techniques of dlfficult generalization. The objective of this work is the classification of couples of bi-stable germs, by means of the local algebras associated with (f1, f2) and and their components, To reach this objective we introduced the notion of cohorent inomorphiom as follows: Let Φ1 : En / If1 + Mk+1n → En / Ig1 + Mk+1n and Φ2 : En / If2 + Mk+1n → En / Ig2 + Mk+1n, be isomorphisms between two algebras associated with the components of the couples (f1, f2, (g1, g2) : Rn, 0 → Rp x Rq, 0. Let us suppose that there are isomorphism θ1 and θ2 of En, for which we have Φ1 (α + If1 + Mk+1n) = θ1 (α) + Ig1 + Mk+1n and Φ2 (α + If2 + Mk+1n) = Φ2 (α) + I,sub>g2 + Mk+1n. We say that isomorphism Φ1 and Φ2 are induced by Phi;1 and Phi;2, respectivaly. (We observe that whenever f K~g then the algebra Qk(f) and Qk(g) are isomorphic vie an induced isomorphical). We say, then, that the isomorphism Φ1 and Φ2 are coherent when they are indiced by the same isomorphism θ : En → En. (We prove that whenever (f1, f2) Bi-K ~(g1, g2 then the algebras Qk(f1) and Qk(g1, Qk (f2) and Qk(g2 are isomorphic according to coehent isomorphism, i.e., isomorhism induced by the only ring-isumorphisms θ : En → En (see chapter IV, 3). Thus the principal theorem can be enunciated: "If bthe couple of germs (F1, f2) and (g1, g2) are bi-stable, then they are bi-A-equivalent if, and only if, the associated algebras are isomorphic through coherent isomorphisms".

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Date de Publication
2019-11-26

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