logo ANNALES DE L'INSTITUT FOURIER

With cedram.org
Table of contents for this issue | Previous article | Next article
Ta Lê Loi
Łojasiewicz inequalities for sets definable in the structure ${\Bbb R}_{{\rm exp}}$
Annales de l'institut Fourier, 45 no. 4 (1995), p. 951-971, doi: 10.5802/aif.1480
Article PDF | Reviews MR 96j:14040 | Zbl 0831.14024 | 1 citation in Cedram

Résumé - Abstract

We consider some variants of Łojasiewicz inequalities for the class of subsets of Euclidean spaces definable from addition, multiplication and exponentiation : Łojasiewicz-type inequalities, global Łojasiewicz inequalities with or without parameters. The rationality of Łojasiewicz’s exponents for this class is also proved.

Bibliography

[1]E. BIERSTONE, Differential functions, Bol. Soc. Bras. Math., vol. 11, n° 2 (1980), 139-190.  MR 83k:58012 |  Zbl 0584.58006
[2]J. BOCHNAK & J. J. RISLER, Sur les exposants de Łojasiewicz, Comment. Math. Helv., 50 (1975), 493-507.  MR 53 #8474 |  Zbl 0321.32006
[3]L. VAN DEN DRIES, Tame topology and 0-minimal structures, mimeographed notes (1991).
[4]L. VAN DEN DRIES & C. MILLER, The field of reals with restricted analytic functions and unrestricted exponentiation : model completeness, 0-minimality, analytic cell decomposition and growth of definable functions, Israel J. Math., 85 (1994), 19-56.  MR 95e:03099 |  Zbl 0823.03017
[5]A. FEKAK, Sur les exposants de Łojasiewicz, Thèse, Rennes (1986).
[6]A. G. KHOVANSKII, On the class of system of transcendental equations, Dokl, Akad. Nauk. SSSR, 255, n° 4 (1980), 804-807 (Russian).  Zbl 0569.32004
[7]A. G. KHOVANSKII, Fewnomials, Transl. Math. Monographs AMS, vol. 88 (1991).  Zbl 0728.12002
[8]J. KNIGHT, A. PILLAY & C. STEINHORN, Definable sets in ordered structures II, Trans. AMS, 295 (1986), 593-605.  MR 88b:03050b |  Zbl 0662.03024
[9]T. L. LOI, Analytic cell decomposition of sets definable in the structure ℝexp, Ann. Pol. Math., LIX3 (1994), 255-266.  Zbl 0806.32001
[10]T. L. LOI, On the global Łojasiewicz inequalities for the class of analytic logarithmico-exponential functions, C. R. Acad. Sci. Paris, t. 318, Série I (1994), 543-548.  MR 95c:32007 |  Zbl 0804.32008
[11]T. L. LOI, Thesis, Krakow (1993).
[12]S. ŁOJASIEWICZ, Ensembles semi-analytiques, I.H.E.S., Bures-sur-Yvette (1965).
[13]B. MALGRANGE, Ideals of differentiable functions, Oxford Univ. Press, London, 1966.  Zbl 0177.17902
[14]M. ROSENLICHT, The rank of a Hardy field, Trans. AMS, 280 (1983), 659-671.  MR 85d:12002 |  Zbl 0536.12015
[15]J. C. TOUGERON, Idéaux de fonctions différentiables, Springer, Berlin, 1972.  MR 55 #13472 |  Zbl 0251.58001
[16]J. C. TOUGERON, Sur certaines algèbres de fonctions analytiques, Séminaire de géométrie algébrique réelle, Paris VII (1986).  MR 89b:32016 |  Zbl 0634.14017
[17]J. C. TOUGERON, Algèbres analytiques topologiquement noethériennes. Théorie de Khovanskii, Ann. Inst. Fourier, Grenoble, 41-4 (1991), 823-840. Cedram |  MR 93f:32005 |  Zbl 0786.32011
[18]J. C. TOUGERON, Inégalités de Łojasiewicz globales, Ann. Inst. Fourier, Grenoble, 41-4 (1991), 841-865. Cedram |  MR 93f:32006 |  Zbl 0748.32007
[19]A. J. WILKIE, Some model completeness results for expansions of the ordered field of real numbers by Pfaffian functions, preprint, Oxford (1991).
[20]A. J. WILKIE, Model completeness results for expansions of the real field II : the exponential function, manuscript, Oxford (1991).
top