In this paper we consider the following problem suggested by T.-C. Kuo. Given a convenient Newton polyhedron and a convergent power series . Under what conditions the topological type of is not affected by perturbations by the functions whose Newton diagram lies above ? If 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 which allows us to define the gradient of with respect to . Using this construction we obtain versions relative to the Newton filtration of Łojasiewicz Inequality for and of Kuiper-Kuo-Paunescu theorem. We show that our result is optimal: if Łojasiewicz Inequality with exponent is not satisfied for then the -jet of with respect to the Newton filtration is not sulficent. In homogeneous case this result is known as Bochnak-Łojasiewicz Theorem. Next we study one parameter families of germs : of analytic functions under the assumption that the leading terms of with respect to the Newton filtration satisfy the uniform Łojasiewicz Inequality. We show that in this case there is a toric modification of such that the family 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 -sufficiency of jets originally due to Takens.
"Polyèdre de Newton et trivialité en famille." J. Math. Soc. Japan 54 (3) 513 - 550, July, 2002. https://doi.org/10.2969/jmsj/1191593907