Finite determinacy of differential forms
type: Item
instance of: scholarly article
title: Finite determinacy of differential forms
zbMATH Open document ID: 0531.58010
DOI: 10.1007/BF01091599
published in: Ukrainian Mathematical Journal
publication date: 1983-01-01T00:00:00Z
review text: In this paper the author explains the problem of finite determinacy of differential 1-forms with respect to the group of changes of the space variables in terms of the properties of the linear approximation for the given form. Let \(K[n]\) be the space of all formal maps \(F: K^n\to K^n\) such that \(F(0)=0\). For any formal 1-form \(w=f_1(x)dx_1+ \ldots +f_n(x)dx_n\) \((f_i(0)=0)\) we associate the map \(F=(f_1,\ldots, f_n)\) from \(K[n]\). (Here, \(K=\mathbb C.)\) Under a change of coordinates \(x=\Phi(y)\), \(\Phi(0)=0\), \(\Phi'(0)=I\), the map \((\Phi')^tF(\Phi)\) corresponds to the new form. We say that these forms are equivalent. We denote \(P_iF\) by the \(i\)-jet of \(F\). We say that \(F\) is \(k\)-determined if every \(H\in K[n]\) such that \(P_kF=P_kH\) is equivalent to \(F\). The author introduces a linear operator \(T(F,s)\) associated with \(F\) and states the following theorem. Theorem 1. Let \(d\) be the maximum of all the numbers \(s\) such that the operator \(T(F,s)\) is not surjective. Then the series \(F\) is either \(d\)-determined or equivalent to \(P_1F\). He states the other two theorems and gives the complete classification of forms in one, two and three variables.
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 58K99
Mathematics Subject Classification ID: 58A17
zbMATH DE Number: 3842804
zbMATH Keywords: finite determinacy
zbMATH Keywords: linear approximation
zbMATH Keywords: formal maps
author: M. Zhitomirskii
reviewed by: Shyuichi Izumiya
MaRDI profile type: Publication
cites work: Stability of \(C^ \infty\) mappings. III: Finitely determined map germs
cites work: Invariant normal forms of formal series
cites work: Q4098934
full work available at URL: https://doi.org/10.1007/bf01091599
OpenAlex ID: W2058756779
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description: scientific article; zbMATH DE number 3842804