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July, 2002 Polyèdre de Newton et trivialité en famille
J. Math. Soc. Japan 54(3): 513-550 (July, 2002). DOI: 10.2969/jmsj/1191593907


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.


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Ould M. ABDERRAHMANE. "Polyèdre de Newton et trivialité en famille." J. Math. Soc. Japan 54 (3) 513 - 550, July, 2002.


Published: July, 2002
First available in Project Euclid: 5 October 2007

zbMATH: 1031.58024
MathSciNet: MR1900955
Digital Object Identifier: 10.2969/jmsj/1191593907

Primary: 14B05
Secondary: 14M25 , 32S45 , 58K45

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

Rights: Copyright © 2002 Mathematical Society of Japan


Vol.54 • No. 3 • July, 2002
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