Abstract
In this paper we investigate the classification of mappings up to \({\mathcal{K}}\)-equivalence. We give several results of this type. We study semialgebraic deformations up to semialgebraic C 0 \({\mathcal{K}}\)-equivalence and bi-Lipschitz \({\mathcal{K}}\)-equivalence. We give an algebraic criterion for bi-Lipschitz \({\mathcal{K}}\)-triviality in terms of semi-integral closure (Theorem 3.5). We also give a new proof of a result of Nishimura: we show that two germs of smooth mappings \({f, g: \mathbb{R}^n \to \mathbb{R}^n}\), finitely determined with respect to \({\mathcal{K}}\)-equivalence are C 0-\({\mathcal{K}}\)-equivalent if and only if they have the same degree in absolute value.
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Acknowledgments
This work was done during visits of MASR to the Université de Provence in Marseille (in May/June 2007) and of GV to the Instituto de Ciências Matemáticas e de Computação in São Carlos (in July 2008). The authors acknowledge the hospitality and the financial support of these institutions during the visits. The first author also acknowledges the financial support from Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Ruas, M.A.S., Valette, G. C0 and bi-Lipschitz \({\mathcal{K}}\) -equivalence of mappings. Math. Z. 269, 293–308 (2011). https://doi.org/10.1007/s00209-010-0728-z
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DOI: https://doi.org/10.1007/s00209-010-0728-z