Abstract
In this work we consider the Paunescu’s singular metric in the source to investigate the bi-Lipschitz and differential sufficiency in the set of weighted jets of map germs with a fixed weighted degree. This approach was motivated by the works of Paunescu, who investigated the V-sufficiency from the weighted point of view considering this metric in the source. Here we show a criterion for the bi-Lipschitz and differentiable sufficiency in the jet space of weighted jets taking into account the effect of this Paunescu’s singular metric on \(\mathbb {R}^n\).
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References
Bochnak, J., Kucharz, W.: Sur les germes d’applications differentiables á singularités isolées. Trans. Am. Math. Soc. 252, 115–131 (1979)
J.C.F. Costa, M.J. Saia, C.H. Soares Junior, Bi-Lipschitz and differentiable sufficiency of weighted jets, J. Sing. 25 (2022), 123–133
L.A. Fávaro, J.C.C. Martins, Sobre a \(C^i\)-suficiência em \(J^r(n,p)\). Métrica - Estudos e pesquisas em Matemática, no. 13, (1981)
N. Kuiper, \(C^1\)-equivalence of functions near isolated critical points. Symp. Inf. Dimens. Topology - Princeton Univ. Press (1972), 199–218
Kuo, T.C.: On \(C^0\)-sufficiency of jets of potential functions. Topology 8, 167–171 (1969)
Kuo, T.C.: Characterization of \(V\)-sufficiency of jets. Topology 11, 115–131 (1972)
Kuo, T.C.: A complete determination of \(C^0\)-sufficiency in \(J^r(2,1)\). Inv. Math. 8, 226–235 (1969)
Lu, Y.C.: Sufficiency of jets in \(J^r(2,1)\) via decomposition. Inv. Math. 10, 119–127 (1970)
Paunescu, L.: \(V\)-sufficiency from the weighted point of view. J. Math. Soc. Jpn. 46(2), 345–354 (1994)
Paunescu, L.: A weighted version of Kuiper-Kuo-Bochnak-Lojaziewicz Theorem. J. Alg. Geom. 2, 69–79 (1993)
Takens, F.: A note on sufficiency of jets. Invent. Math. 13, 225–231 (1971)
Funding
Funding was provided by Fundação de Amparo à Pesquisa do Estado de São Paulo (2019/21181-0, 2023/08176-2, 2019/21181-0).
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Costa, J.C.F., Saia, M.J. & Soares Junior, C.H. Sufficiency of weighted jets using Paunescu’s singular metric. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 59 (2024). https://doi.org/10.1007/s13398-024-01553-8
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DOI: https://doi.org/10.1007/s13398-024-01553-8