×

zbMATH — the first resource for mathematics

Density of semi-pfaffian sets. (Densité des ensembles semi-pfaffiens.) (French) Zbl 0917.32003
Semi-pfaffian sets introduced by R. Moussu in 1988, following and idea of Hovansky, has developed ever since, mostly in Dijon (Moussu, Rocke, Lion, Rolin), forming a very useful theory. In ordinary differential equations, even if the equation is analytic, the solutions are very rarely subanalytic but they are mostly semi-pfaffian [cf. R. Moussu and C.-A. Roche, Invent. Math. 105, No. 2, 431-441 (1991; Zbl 0769.58050)].
In the present paper the author proves the existence of densities in each boundary point of a semi-pfaffian set, generalizing therefore the results of Kurdyka-Poly-Raby [K. Kurdyka, J.-B. Poly and G. Raby, Lect. Notes Math. 1420, 170-177 (1990; Zbl 0694.32001); K. Kurdyka and G. Raby, Inst. Fourier 39, No. 3, 753-771 (1989; Zbl 0673.32015)] about subanalytic sets, as well as these of Roche [C.-A. Roche, Astérisque 222, 373-387 (1994; Zbl 0831.32004)].
The methods used are these of integral geometry (cf. the book [L. A. Santaló, ‘Integral geometry and geometric probability’ (1976; Zbl 0342.53049)] and the work of Langevin [R. Langevin, ‘Un peu de géométrie intégrale, Images des Mathématiques, CNRS, 58-67 (1995)]), namely the Cauchy-Crofton integration formula, and adaption of the subanalytic proof.
The paper is very clear, well written, and an exhaustive bibliography makes it easy to follow the author’s thought.

MSC:
32C05 Real-analytic manifolds, real-analytic spaces
32C25 Analytic subsets and submanifolds
32B20 Semi-analytic sets, subanalytic sets, and generalizations
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] Denkowska ( S. ) et Stasica ( J. ) .- Ensembles sous-analytiques à la polonaise , preprint ( 1985 ). · Zbl 0584.32013
[2] Khovanskii ( A.G. ) . - Real analytic varieties with finiteness property and complex abelian integrals , Funct. Anal. and Appl. 18 ( 1984 ), pp. 119 - 127 . MR 745698 | Zbl 0584.32016 · Zbl 0584.32016
[3] Kurdyka ,( K. ), Poly ( J.-B. ) et Raby ( G. ) .- Moyennes des fonctions sous-analytiques, densité, cône tangent , L.N.M. 1420 ( proceedings Trento 1988 ), pp. 170 - 177 . MR 1051211 | Zbl 0694.32001 · Zbl 0694.32001
[4] Kurdyka ( K. ) et Raby ( G. ) .- Densité des ensembles sous-analytiques , Ann. Institut Fourier 39 , n^\circ 3 ( 1989 ), pp. 753 - 771 . Numdam | MR 1030848 | Zbl 0673.32015 · Zbl 0673.32015
[5] Langevin ( R. ) .- Un peu de géométrie intégrale , Images des Mathématiques , CNRS ( 1995 ), pp. 58 - 67 .
[6] Lelong ( P. ) . - Intégration sur un ensemble analytique complexe , Bull. Soc. math. France 85 ( 1957 ), pp. 239 - 262 . Numdam | MR 95967 | Zbl 0079.30901 · Zbl 0079.30901
[7] Lion ( J.-M. ) et Roche ( C.-A. ) .- Topologie des hypersurfaces pfaffiennes , Bull. Soc. math. France , 124 ( 1996 ), pp. 35 - 59 . Numdam | MR 1395006 | Zbl 0852.32007 · Zbl 0852.32007
[8] Lion ( J.-M. ) et Rolin ( J.-P. ) .- Homologie des ensembles semi-pfaffiens , Ann. Institut Fourier 46 , n^\circ 3 ( 1996 ), pp. 723 - 741 . Numdam | MR 1411726 | Zbl 0853.32004 · Zbl 0853.32004
[9] Lion ( J.-M. ) .- Partitions normales de Lojasiewicz et hypersurfaces pfaffiennes , C.R.A.S. de Paris , Série I , 311 ( 1990 ), pp. 453 - 456 . MR 1075669 | Zbl 0715.32002 · Zbl 0715.32002
[10] Lion ( J.-M. ) .- Étude des hypersurfaces pfaffiennes , Thèse de l’Université de Bourgogne , novembre 1991 .
[11] Lojasiewicz ( S. ) .- Ensembles semi-analytiques , Preprint I.H.E.S. ( 1965 ).
[12] Moussu ( R. ) et Roche ( C.-A. ) .- Théorie de Hovanskii et problème de Dulac , Invent. Math. 105 ( 1991 ), pp. 431 - 441 . MR 1115550 | Zbl 0769.58050 · Zbl 0769.58050
[13] Moussu ( R. ) et Roche ( C.-A. ) .- Théorèmes de finitude uniforme pour les variétés pfaffiennes de Rolle , Ann. Institut Fourier 42 , n^\circ 1 -2 ( 1992 ), pp. 393 - 420 . Numdam | MR 1162568 | Zbl 0759.32005 · Zbl 0759.32005
[14] Roche ( C.-A. ) .- Densities for certain leaves of real analytic foliations , Astérisque 222 ( 1994 ), pp. 373 - 387 . MR 1285396 | Zbl 0831.32004 · Zbl 0831.32004
[15] Santaló ( L.A. ) . - Integral geometry and geometric probability dans ”Encyclopedia of mathematics and its applications ”, Addison-Wesley , Reading , 1 ( 1976 ), MR 433364 | Zbl 0342.53049 · Zbl 0342.53049
[16] Vitushkin ( A.G. ) .- Variation multidimensionnelle , Go. Tekh. Izdat Moscou , 1955 (en russe).
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.