## Journal of the Mathematical Society of Japan

### Polyèdre de Newton et trivialité en famille

Ould M. ABDERRAHMANE

#### 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 $f_{t}$ : $(R^{n},0)\rightarrow(R,0)$ 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.

#### Article information

Source
J. Math. Soc. Japan, Volume 54, Number 3 (2002), 513-550.

Dates
First available in Project Euclid: 5 October 2007

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1191593907

Digital Object Identifier
doi:10.2969/jmsj/1191593907

Mathematical Reviews number (MathSciNet)
MR1900955

Zentralblatt MATH identifier
1031.58024

#### Citation

M. ABDERRAHMANE, Ould. Polyèdre de Newton et trivialité en famille. J. Math. Soc. Japan 54 (2002), no. 3, 513--550. doi:10.2969/jmsj/1191593907. https://projecteuclid.org/euclid.jmsj/1191593907