Multiplicity of iterated Jacobian extensions of weighted homogeneous map germs
type: Item
instance of: scholarly article
title: Multiplicity of iterated Jacobian extensions of weighted homogeneous map germs
zbMATH Open document ID: 0983.32029
published in: Hokkaido Mathematical Journal
publication date: 2002-04-23T00:00:00Z
review text: The authors define an invariant \(e_{\mathbf i}(f)\) called the algebraic multiplicity of the analytic map germ \(f:(\mathbb{C}^n,0) \to(\mathbb{C}^p,0)\) with respect to the Boardman symbol \({\mathbf i}=(i_1,\dots,i_k)\). This invariant is the multiplicity (in the Hilbert-Samuel sense) of the local ring \({\mathcal O}_n/J_{\mathbf i}(f)\) where \(J_{\mathbf i}(f)\) is an iterated jacobian ideal defined by \(f\), it generalizes other well-known invariants like the Milnor number. When \({\mathcal O}_n/J_{\mathbf i}(f)\) has dimension 0, \(e_{ \mathbf i}(f)\) is the colength of \(J_{\mathbf i}(f)\) in \({\mathcal O}_n\) and coincides with the invariant previously defined by the second author and \textit{M. J. Saia} in [Glasg. Math. J. 40, No. 1, 21-32 (1998; Zbl 0919.58010)]. They give some formulae for this invariant when \(f\) is weighted homogeneous. The computation is based on some results on the multiplicity of determinantal rings that they prove using the Eagon-Northcott complex.
Mathematics Subject Classification ID: 32S10
Mathematics Subject Classification ID: 32S30
zbMATH DE Number: 1532964
zbMATH Keywords: algebraic multiplicity
zbMATH Keywords: analytic map germ
zbMATH Keywords: iterated jacobian ideal
zbMATH Keywords: Milnor number
author: Carles Bivià-Ausina
author: J. J. Nuño-Ballesteros
reviewed by: P-A Mazet
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1350912976
OpenAlex ID: W2043200440
DOI: 10.14492/HOKMJ/1350912976
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description: scientific article; zbMATH DE number 1532964