Finite determinacy of differential forms
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.
scientific article; zbMATH DE number 3842804
scientific article