Polyèdre de Newton et trivialité en famille

Abstract

In this paper 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 $C^0$ sulficent. In homogeneous case this result is known as Bochnak-Łojasiewicz Theorem. Next we study one parameter families of germs $f_t$ : $(R^n,0)\to (R,0)$ of analytic functions under the assumption that the leading terms of $f_t$ with respect to the Newton filtration satisfy the uniform Łojasiewicz Inequality. We show that in this case there is a toric modification $\pi$ of $R^n$ such that the family $f_t\circ \pi$ is analytically trivial. Our result implies in particular the criteria for blow-analytic trivliality 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.