×

Intégration des fonctions sous-analytiques et volumes des sous-ensembles sous-analytiques. (Integration of subanalytic functions and volumes of subanalytic subspaces). (French) Zbl 0912.32007

The work speaks about the integration of global subanalytic functions and seems very useful for the calculus. As the authors say, it was inspired by L. van den Dries and Rémi Langevin, both outstanding mathematicians with many great ideas. The proofs use in a very interesting way the preparation theorem by the authors [J.-M. Lion and J.-P. Rolin, Ann. Inst. Fourier 47, No. 3, 859-884 (1997; Zbl 0873.32004)] and a very useful, unfortunately not often used, work by K. Kurdyka and G. Raby [Ann. Inst. Fourier 39, No. 3, 753-771 (1989; Zbl 0673.32015)] about the density of subanalytic sets.
It is a pity that the Introduction (I) is dispersed and misleading, with no order in the ideas, quotations and an overly optimistic statement about the possibility of obtention of algebras stable by integration. The reviewer is tempted to advise a beginner reader to skip the introduction and read the text first, as well as not to be impressed by the bibliography, most of which is not used except in the introduction.

MSC:

32B20 Semi-analytic sets, subanalytic sets, and generalizations
32B15 Analytic subsets of affine space

References:

[1] [AGV] , et , Singularités des applications différentiables, MIR, Moscou, 1986.
[2] [Ba] , Développement asymptotique des fonctions obtenues par intégration sur les fibres, Inv. Math., 68 (1982), 129-174. · Zbl 0508.32003
[3] [BM] et , Semianalytic and subanalytic sets, Publ. Math. IHES, 67 (1988), 5-42. · Zbl 0674.32002
[4] [DMM] , et , The elementary theory of restricted analytic fields with exponentiation, Annals of Maths, 140 (1994), 183-205. · Zbl 0837.12006
[5] [Ga] , Projections of semi-analytic sets, Funct. Anal. Appl., 2 (1968), 282-291. · Zbl 0179.08503
[6] [Hi] , Subanalytic sets, Number Theory, Algebraic Geometry and Commutative Algebra, Tokyo, Kinokuniya, (1973), 453-493. · Zbl 0297.32008
[7] [Je] , Intégration sur les fibres d’une fonction analytique, dans Introduction à la théorie algébrique des systèmes différentiels, 1-39, Travaux en cours, 34, Hermann, Paris (1988). · Zbl 0676.32001
[8] [KP] et , Volume of small extrinsic ball in a submanifold, Bull. London Math. Soc., 21 (1989), 87-92. · Zbl 0641.53007
[9] [KR] et , Densité des ensembles sous-analytiques, Ann. Inst. Fourier, 39-3 (1989), 753-771. · Zbl 0673.32015
[10] [Fl1] , Sequences of rational torsion on abelian varieties, Inventiones Math., 106 (1991
[11] [LDT] , Geometry of monodromy and Nilpotency exponent, manuscrit (1979).
[12] [Le] , Intégration sur un ensemble analytique complexe, Bull. Soc. Math. France, 85 (1957), 239-262. · Zbl 0079.30901
[13] [LR] et , Théorème de préparation pour les fonctions logarithmico-exponentielles, Ann. Inst. Fourier, 47-3 (1997), 859-884. · Zbl 0873.32004
[14] [Lo] , Volumes des tubes autour des singularités, Duke Math. Journal, 53 (1986), 443-455. · Zbl 0653.32006
[15] [Łoj] , Stratifications et triangulations sous-analytiques, Università degli Studi di Bologna (1986). · Zbl 0617.32011
[16] [Ma] , Intégrales asymptotiques et monodromie, Ann. Scien. ENS, 7 (1974), 405-430. · Zbl 0305.32008
[17] [Mi] , Expansions of the real field with power functions, Ann. Pure Appl. Logic, 68 (1994). · Zbl 0823.03018
[18] [Ni] , dans Arkiv för Mathematik.
[19] [Pa] , Lipschitz stratification of subanalytic sets, Ann. Scient. ENS, 27 (1994), 661-996. · Zbl 0819.32007
[20] [Ro] , Densities for certain leaves of real analytic foliations, Astérisque, 222 (1994), 373-387. · Zbl 0831.32004
[21] [Sa] , Integral geometry and geometric probability dans Encyclopedia of mathematics and its applications, Addison-Wesley, Reading, Vol 1. · Zbl 0342.53049
[22] [To] , Paramétrisations de petits chemins en géométrie analytique réelle, preprint Université de Rennes. · Zbl 0852.32006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.