Journal of the Mathematical Society of Japan

Polyèdre de Newton et trivialité en famille


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In this paper we consider the following problem suggested by T.-C. Kuo. Given a convenient Newton polyhedron Γ 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 Γ? If Γ 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 Γ which allows us to define the gradient of f with respect to Γ. 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 C0 sulficent. In homogeneous case this result is known as Bochnak-Łojasiewicz Theorem. Next we study one parameter families of germs ft : (Rn,0)(R,0) of analytic functions under the assumption that the leading terms of ft with respect to the Newton filtration satisfy the uniform Łojasiewicz Inequality. We show that in this case there is a toric modification π of Rn such that the family ftπ 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 Ck-sufficiency of jets originally due to Takens.

Article information

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

First available in Project Euclid: 5 October 2007

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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]

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


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.

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