Finiteness property for generalized abelian integrals
Annales de l'Institut Fourier, Volume 53 (2003) no. 3, p. 767-785
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.
DOI : https://doi.org/10.5802/aif.1959
Classification:  32B15,  32B20
Keywords: abelian integrals, preparation theorem, o-minimal structures, diophantine conditions
@article{AIF_2003__53_3_767_0,
     author = {Soufflet, R\'emi},
     title = {Finiteness property for generalized abelian integrals},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {53},
     number = {3},
     year = {2003},
     pages = {767-785},
     doi = {10.5802/aif.1959},
     zbl = {1034.32007},
     mrnumber = {2008440},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2003__53_3_767_0}
}
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. http://www.numdam.org/item/AIF_2003__53_3_767_0/

[Ar] V. Arnold Chapitres supplémentaires de la théorie des équations différentielles ordinaires, Mir (1980) | MR 626685 | Zbl 0455.34001

[BCR] J. Bochnak; M. Coste; F. Roy Géométrie algébrique réelle, Springer-Verlag (1987) | MR 949442 | Zbl 0633.14016

[BG] M. Berger; B. Gostiaux Géométrie différentielle : variétés, courbes et surfaces, Presses Universitaires de France (1987) | MR 903026 | Zbl 0619.53001

[BM] E. Bierstone; P. Milman Semianalytic and subanalytic sets, Publ. Math. IHES, Tome 67 (1988), pp. 5-42 | Numdam | MR 972342 | Zbl 0674.32002

[BR] R. Benedetti; J.-J. Risler Real algebraic and semialgebraic sets, Hermann (1990) | MR 1070358 | Zbl 0694.14006

[CY] G. Comte; Y. Yomdin A course on metric properties of algebraic sets (2000) (preprint)

[DM] L. Van Den Dries; C. Miller Geometric categories and o-minimal structure, Duke Math. J, Tome 84 (1996) no. 2 | Article | MR 1404337 | Zbl 0889.03025

[DMM] L. Van Den Dries; A. Macintyre; D. Marker The elementary theory of restricted analytic fields with exponentiation, Ann. of Maths, Tome 140 (1994), pp. 183-205 | Article | MR 1289495 | Zbl 0837.12006

[Du] H. Dulac Sur les cycles limites, Bull. Soc. Math. Fr, Tome 51 (1923), pp. 45-188 | JFM 49.0304.01 | Numdam | MR 1504823

[Ga] A.M. Gabrielov Projections of semi-analytic sets, Funct. Anal. Appl, Tome 2 (1968), pp. 282-291 | Article | MR 245831 | Zbl 0179.08503

[GS] D.Y. Grigoriev; M.F. Singer Solving ordinary differential equations in terms of series with real exponents, Transactions Amer. Math. Soc, Tome 327 (1991) no. 1, pp. 329-351 | Article | MR 1012519 | Zbl 0758.12004

[Kh1] A.G. Khovanskii Real analytic varieties with the finitness property and complex abelian integrals, Funct. Anal. and Appl, Tome 18 (1984), pp. 119-127 | Article | MR 745698 | Zbl 0584.32016

[Kh2] A.G. Khovanskii Fewnomials, A.M.S., mathematical monographs, Tome 88 (1991) | MR 1108621 | Zbl 0728.12002

[Lo] S. Łojasiewicz On semianalytic and subanalytic geometry, Banach Center Publication, Tome 34 (1995), pp. 89-104 | Zbl 0841.32003

[LR1] J.-M. Lion; J.-P. Rolin Théorème de préparation pour les fonctions logarithmico-exponentielles, Ann. Inst. Fourier, Tome 47 (1997) no. 3, pp. 859-884 | Article | Numdam | MR 1465789 | Zbl 0873.32004

[LR2] J.-M. Lion; J.-P. Rolin Intégration des fonctions sous-analytiques et volume des sous-analytiques, Ann. Inst. Fourier, Tome 48 (1998) no. 3, pp. 755-767 | Article | Numdam | MR 1644093 | Zbl 0912.32007

[Mi] C. Miller Expansions of the real field with power functions, Ann. Pure Appl. Logic, Tome 68 (1994) | MR 1278550 | Zbl 0823.03018

[Mo] R. Moussu Le problème de la finitude du nombre de cycles limites, Séminaire Bourbaki (1985) (Astérisque) Tome 655 (1987), p. 145-146 | Numdam | Zbl 0617.58028

[MR] R. Moussu; C. Roche Théorie de Hovanskii et problème de Dulac, Invent. Math, Tome 105 (1991) no. 2, pp. 431-441 | Article | MR 1115550 | Zbl 0769.58050

[Pa] A. Parusiński Lipschitz stratification of subanalytic sets, Ann. Scient. École Normale Supérieure, 4e série, Tome 27 (1994), pp. 661-696 | Numdam | MR 1307677 | Zbl 0819.32007

[Sa] M. Saavedra Développement asymptotique de la fonction période, CRAS, Tome 319 (1994), pp. 563-566 | MR 1298283 | Zbl 0814.34038

[So1] R. Soufflet Propriétés oscillatoires des intégrales de \x-fonctions, CRAS, Tome 333 (2001), pp. 461-464 | MR 1859237 | Zbl 1044.32002

[So2] R. Soufflet Asymptotic expansions of logarithmic-exponential functions, Bull. Braz. Math. Soc., New Series, Tome 33 (2002) no. 1, pp. 125-146 | MR 1934286 | Zbl 1027.32013

[Ta] A. Tarski A decision method for elementary algebra and geometry, University of California Press, Berkeley and Los Angeles, Calif. (1951) | MR 44472 | Zbl 0044.25102

[To] J.-C. Tougeron Paramétrisations de petits chemins en géométrie analytique réelle, Singularities and differential equations. Proceedings of a symposium, Warsaw, Banach Cent. Publ, Tome 33 (1996), pp. 421-436 | Zbl 0852.32006

[Va] A.N. Varchenko Estimate of the number of zeros of an abelian integral depending an a parameter and limit cycles, Funct. Anal. and Appl, Tome 18 (1984), pp. 98-107 | Article | MR 745696 | Zbl 0578.58035

[Wi] A.J. Wilkie Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc, Tome 9 (1996) no. 4, pp. 1051-1094 | Article | MR 1398816 | Zbl 0892.03013

[Yo] Y. Yomdin Metric properties of semialgebraic sets and mappings and their applications in smooth analysis, Géométrie réelle, Systèmes différentiels et théorie de Hodge. Travaux en cours, Hermann, Paris, Tome 24 | Zbl 0632.58009