Abstract
Let \(f,g:({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^m,0)\) be \(C^{r+1}\) mappings and let \(Z=\{x\in \mathbf {\mathbb {R}}^n:\nu (df (x))=0\}\), \(0\in Z\), \(m\le n\). We will show that if there exist a neighbourhood U of \(0\in {\mathbb {R}}^n\) and constants \(C,C'>0\) and \(k>1\) such that for \(x\in U\)
for any \(i\in \{1,\dots , m\}\) and for any \(s \in \mathbf {\mathbb {N}}^n_0\) such that \(|s|\le r\), then there exists a \(C^r\) diffeomorphism \(\varphi :({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^n,0)\) such that \(f=g\circ \varphi \) in a neighbourhood of \(0\in {\mathbb {R}}^n\). By \(\nu (df)\) we denote the Rabier function.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bekka, K., Koike, S.: Characterisation of \(V\)-sufficiency and \(C^0\)-sufficiency of relative jets. https://arxiv.org/pdf/1703.07069.pdf
Bochnak, J.: Relévement des jets. In: Séminaire Pierre Lelong (Analyse), (année 1970–1971). Lecture Notes in Mathematics, vol. 275, pp. 106–118. Springer, Berlin (1972)
Jelonek, Z.: On the generalized critical values of a polynomial mapping. Manuscr. Math. 110, 145–157 (2003)
Kuiper, N.H.: \(C^1\)-equivalence of functions near isolated critical points. In: Symposium on Infinite Dimensional Topology (Louisiana State University, Baton Bouge, 1967). Annals of Mathematics Studies, vol. 69, pp. 199–218. Princeton University Press, Princeton (1972)
Kuo, T.C.: On \(C^0\)-sufficiency of jets of potential functions. Topology 8, 167–171 (1969)
Kurdyka, K., Orro, P., Simon, S.: Semialgebraic Sard theorem for generalized critical values. J. Differ. Geom. 56, 67–92 (2000)
Migus, P.: \(C^r\)-right equivalence of analytic functions. Demonstr. Math. 48(2), 313–321 (2015)
Migus, P.: Local \(C^r\)-right equivalence of \(C^{r+1}\) functions. Glasg. Math. J. 59(1), 265–272 (2017)
Migus, P., Rodak, T., Spodzieja, S.: Finite determinancy of non-isolated singularities. Ann. Pol. Math. 117(3), 197–206 (2016)
Rabier, P.J.: Ehresmann fibrations and Palais–Smale conditions for morphisms of Finsler manifolds. Ann. Math. Second Ser. 146, 647–691 (1997)
Rodak, T., Spodzieja, S.: Equivalence of mappings at infinity. Bull. Sci. Math. 136(6), 679–686 (2012)
Takens, F.: A note on sufficiency of jets. Invent. Math. 13, 225–231 (1971)
Tougeron, J.C.: Idéaux de fonctions différentiables. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 71. Springer, New York (1972)
Xu, X.: \(C^0\)-sufficiency, Kuiper–Kuo and Thom conditions for non-isolated singularity. Acta Math. Sin. (Engl. Ser.) 23(7), 1251–1256 (2007)
Acknowledgements
I am deeply grateful to Mihai Tibăr for his valuable comments and advices.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This research was partially supported by NCN, Grant Number 2015/17/B/ST1/02637.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Migus, P. Sufficient conditions for local equivalence of mappings. Arch. Math. 112, 395–405 (2019). https://doi.org/10.1007/s00013-018-1289-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-018-1289-3