## Journal of the Mathematical Society of Japan

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

### 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.

#### 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

**Subjects**

Primary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx]

Secondary: 58K45: Singularities of vector fields, topological aspects 14M25: Toric varieties, Newton polyhedra [See also 52B20] 32S45: Modifications; resolution of singularities [See also 14E15]

**Keywords**

Newton polyhedron $C^{0}$-sufficiency modified analytic trivialization toric modification

#### 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