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Infinite determinacy on a closed set for smooth germs with non-isolated singularities. (English) Zbl 1090.58022
The paper deals with infinite determinacy of germs of \(C^\infty\) functions, \(f:({\mathbb{R}}^n,0) \rightarrow ({\mathbb{R}},0)\). Let \(\psi_1, \dots, \psi_p\) (\(p\leq n\)) be germs of analytic functions, \(\psi_:({\mathbb{R}}^n,0) \rightarrow ({\mathbb{R}},0)\), I the ideal generated by \(\psi_1, \dots , \psi_p\) and \(X\) the germ of the zero set, \(X=\{x: \psi_1(x)=\dots =\psi_p(x)=0 \}\). The only assumption is that the complement of the singular set \(\Sigma =\{x \in X:d\psi_(x)\wedge \dots \wedge d\psi_p(x)=0\}\) is dense in \(X\). The author restricts himself to the germs of \(C^\infty\) functions of the form \(f(x)=\sum_{i,j=1}^{p} f_{i,j}(x)\psi_i(x)\psi_j(x)\), where \(f_{i,j}=f_{j,i}\). Let \(Y\) be a germ at the origin of a closed subset of \({\mathbb{R}}^n\), containing \(\Sigma\).
The main result of the paper is the proof that the following conditions are equivalent:
1. The germ \(f\) is infinitely determined relative to \(Y\), i.e., for any germ \(u \in m_Y^\infty I^2\) (\(m_Y^\infty\) denotes the ideal of germs flat at \(Y\)), there exists a germ of a diffeomorphism \(\Phi\) such that \(f(x)+u(x)=f(\Phi(x))\), and for \(x\in X \setminus Y\) \(\Phi(x)=x\).
2. \(| | \nabla f| | \) and \(\det(f_{i,j})\) satisfy the Łojasiewicz inequalities, i.e., there are such positive \(C\) and \(\alpha\) that close to the origin \(| | \nabla f (x)| | \geq C\, \text{dist}(x,X\cup Y)^\alpha\), for \(x \in {\mathbb{R}}^n\), and \(| \det(f_{i,j}(x)| \geq C\, \text{dist}(x,Y)^\alpha \), for \(x \in X\).

MSC:
58K40 Classification; finite determinacy of map germs
32S05 Local complex singularities
26E10 \(C^\infty\)-functions, quasi-analytic functions
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