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
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.
J. Math. Soc. Japan, Volume 54, Number 3 (2002), 513-550.
First available in Project Euclid: 5 October 2007
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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