Finiteness property for generalized abelian integrals

Soufflet, Rémi

doi:10.5802/aif.1959

Soufflet, Rémi ¹

¹ Institute of Mathematics UJ, ul. Reymonta 4, 30-059 Kraków (Pologne)

Annales de l'Institut Fourier, Volume 53 (2003) no. 3, pp. 767-785.

Abstract
Résumé

We study the integrals of real functions which are finite compositions of globally subanalytic maps and real power functions. These functions have finiteness properties very similar to those of subanalytic functions. Our aim is to investigate how such finiteness properties can remain when taking the integrals of such functions. The main result is that for almost all power maps arising in a $x^{λ}$ -function, its integration leads to a non-oscillating function. This can be seen as a generalization of Varchenko and Khovanskii’s finiteness theorems for abelian integrals.

Cette étude porte sur les intégrales de fonctions réelles qui sont des composées finies d'applications sous-analytiques globales et de fonctions puissances à exposants réels. Ces dernières possèdent des propriétés de finitude semblables à celles des fonctions sous- analytiques. Nous montrons que pour presque toutes valeurs des exposants intervenant dans la définition d'une telle fonction, son intégrale sur les fibres d'une fonction du même type est non-oscillante. Ce résultat peut se voir comme une généralisation des théorèmes de finitude des zéros des intégrales abéliennes de Varchenko et Khovanskii.

MR: 2008440 | Zbl: 1034.32007

DOI: 10.5802/aif.1959

Classification: 32B15, 32B20
Keywords: abelian integrals, preparation theorem, o-minimal structures, diophantine conditions

Author's affiliations:

Soufflet, Rémi ¹

¹ Institute of Mathematics UJ, ul. Reymonta 4, 30-059 Kraków (Pologne)

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Soufflet, Rémi. Finiteness property for generalized abelian integrals. Annales de l'Institut Fourier, Volume 53 (2003) no. 3, pp. 767-785. doi : 10.5802/aif.1959. https://aif.centre-mersenne.org/articles/10.5802/aif.1959/

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