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Newton polyhedron and trivialization ‘en famille’. (Polyèdre de Newton et trivialité en famille.) (French) Zbl 1031.58024

Summary: We consider the following problem suggested by T.-C. Kuo. Given a convenient Newton polyhedron \(\Gamma\) and a convergent power series \(f\). Under what conditions the topological type of \(f\) is not affected by perturbations by the functions whose Newton diagram lies above \(\Gamma\)? If \(\Gamma\) consists of one face only (weighted homogeneous case) then the answer is given by theorems of Kuiper-Kuo and of Paunescu. In order to answer this problem we introduce a pseudo-metric adapted to the polyhedron \(\Gamma\) which allows us to define the gradient of \(f\) with respect to \(\Gamma\). Using this construction we obtain versions relative to the Newton filtration of Łojasiewicz inequality for \(f\) and of Kuiper-Kuo-Paunescu theorem. We show that our result is optimal: if Łojasiewicz inequality with exponent \(r\) is not satisfied for \(f\) then the \(r\)-jet of \(f\) with respect to the Newton filtration is not \(C^0\) sufficent. In homogeneous case this result is known as Bochnak-Łojasiewicz theorem.
Next we study one parameter families of germs \(f_t: (\mathbb{R}^n, 0)\to (\mathbb{R},0)\) of analytic functions under the assumption that the leading terms of \(f_1\) with respect to the Newton filtration satisfy the uniform Łojasiewicz inequality. We show that in this case there is a toric modification \(\pi\) of \(\mathbb{R}^n\) such that the family \(f_t\circ \pi\) is analytically trivial. Our result implies in particular the criteria for blow-analytic triviality due to Kuo, Fukui-Paunescu, and Fukui-Yoshinaga.
Our technique can be also used to improve the criteria on \(C^k\)-sufficiency of jets originally due to Takens.

MSC:

58K40 Classification; finite determinacy of map germs
32S05 Local complex singularities
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