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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
##### Keywords:
singularites; infinite determinacy
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##### References:
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