\(C^\ell\)-determinacy of weighted homogeneous germs
type: Item
instance of: scholarly article
title: \(C^\ell\)-determinacy of weighted homogeneous germs
zbMATH Open document ID: 0873.58010
author: Maria Aparecida Soares Ruas
author: Marcelo José Saia
published in: Hokkaido Mathematical Journal
publication date: 1997-04-07T00:00:00Z
review text: In the paper the notion of \({\mathcal C}^\ell\)-determinacy relative to the standard Mather's groups \(\mathcal R, C\) and \(\mathcal K\) is considered. The authors generalize results of \textit{M. A. S. Ruas} [Math. Scand. 59, No. 1, 59-70 (1986; Zbl 0632.58013)] and determine explicitly the degree of \({\mathcal C}^\ell\)-determinacy of weighted homogeneous map-germs satisfying some kind of Lojasiewicz condition. As an application they obtain equisingular deformations in the well-known Briançon-Speder example [\textit{J. Briançon} and \textit{J. N.-P. Speder}, C. R. Acad. Sci., Paris, Sér. A, 280 365-367 (1975; Zbl 0331.32010)] and prove that the corresponding family \(F(x,y,z,t) = z^5 + xy^7 + x^{15} + ty^6z\) defined in \({\mathbb{R}}^4\) is non-equisingular. The authors also remark that similar results are contained in an earlier work [\textit{S. Bromberg} and \textit{S. Lopez de Medrano}, Res. Notes Math. Ser. 333, 179-188 (1995; Zbl 0839.58011)].
reviewed by: Aleksandr G. Aleksandrov
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 58K99
zbMATH DE Number: 992415
zbMATH Keywords: \({\mathcal C}^ \ell\)-determinacy
zbMATH Keywords: weighted homogeneous map-germs
zbMATH Keywords: Lojasiewicz condition
zbMATH Keywords: perturbations
zbMATH Keywords: controlled vector fields
zbMATH Keywords: control functions
zbMATH Keywords: equisingularity
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1351257806
OpenAlex ID: W2039400188
DOI: 10.14492/HOKMJ/1351257806
description: scientific article; zbMATH DE number 992415