Book Review
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Book Information:
Author:
Lou van den Dries
Title:
Tame topology and o-minimal structures
Additional book information:
Cambridge Univ. Press,
New York,
1998,
x + 180 pp.,
ISBN 0-521-59838-9,
$39.95$
Riccardo Benedetti and Jean-Jacques Risler, Real algebraic and semi-algebraic sets, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1990. MR 1070358
Edward Bierstone and Pierre D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 5–42. MR 972342
Jacek Bochnak, Michel Coste, and Marie-Françoise Roy, Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 36, Springer-Verlag, Berlin, 1998. Translated from the 1987 French original; Revised by the authors. MR 1659509, DOI 10.1007/978-3-662-03718-8
Lou van den Dries, Remarks on Tarski’s problem concerning $(\textbf {R},\,+,\,\cdot ,\,\textrm {exp})$, Logic colloquium ’82 (Florence, 1982) Stud. Logic Found. Math., vol. 112, North-Holland, Amsterdam, 1984, pp. 97–121. MR 762106, DOI 10.1016/S0049-237X(08)71811-1
Lou van den Dries, A generalization of the Tarski-Seidenberg theorem, and some nondefinability results, Bull. Amer. Math. Soc. (N.S.) 15 (1986), no. 2, 189–193. MR 854552, DOI 10.1090/S0273-0979-1986-15468-6
Lou van den Dries, Angus Macintyre, and David Marker, The elementary theory of restricted analytic fields with exponentiation, Ann. of Math. (2) 140 (1994), no. 1, 183–205. MR 1289495, DOI 10.2307/2118545
Lou van den Dries and Chris Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), no. 2, 497–540. MR 1404337, DOI 10.1215/S0012-7094-96-08416-1
Lou van den Dries and Patrick Speissegger, The real field with convergent generalized power series, Trans. Amer. Math. Soc. 350 (1998), no. 11, 4377–4421. MR 1458313, DOI 10.1090/S0002-9947-98-02105-9
[9] L. van den Dries and P. Speissegger, The field of reals with multisummable series and the exponential function, Proc. London Math. Soc., to appear.
Alexandre Grothendieck, Esquisse d’un programme, Geometric Galois actions, 1, London Math. Soc. Lecture Note Ser., vol. 242, Cambridge Univ. Press, Cambridge, 1997, pp. 5–48 (French, with French summary). With an English translation on pp. 243–283. MR 1483107
A. G. Hovanskiĭ, A class of systems of transcendental equations, Dokl. Akad. Nauk SSSR 255 (1980), no. 4, 804–807 (Russian). MR 600749
Anand Pillay and Charles Steinhorn, Definable sets in ordered structures. I, Trans. Amer. Math. Soc. 295 (1986), no. 2, 565–592. MR 833697, DOI 10.1090/S0002-9947-1986-0833697-X
Ya’acov Peterzil and Sergei Starchenko, A trichotomy theorem for o-minimal structures, Proc. London Math. Soc. (3) 77 (1998), no. 3, 481–523. MR 1643405, DOI 10.1112/S0024611598000549
[14] Y. Peterzil, A. Pillay and S. Starchenko, Simple algebraic and semialgebraic groups over real closed fields, Trans. AMS, to appear.
Anand Pillay, On groups and fields definable in $o$-minimal structures, J. Pure Appl. Algebra 53 (1988), no. 3, 239–255. MR 961362, DOI 10.1016/0022-4049(88)90125-9
Anand Pillay and Charles Steinhorn, Definable sets in ordered structures. I, Trans. Amer. Math. Soc. 295 (1986), no. 2, 565–592. MR 833697, DOI 10.1090/S0002-9947-1986-0833697-X
Leila Schneps and Pierre Lochak (eds.), Geometric Galois actions. 1, London Mathematical Society Lecture Note Series, vol. 242, Cambridge University Press, Cambridge, 1997. Around Grothendieck’s “Esquisse d’un programme”. MR 1483106, DOI 10.1017/CBO9780511758874
[18] P. Speissegger, The Pfaffian closure of an o-minimal structure, J. Reine Angew. Math. 508 (1999), 198-211. CMP 99:09
[19] B. Tessier, Tame and stratified objects, in [17], 231-243.
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. 9 (1996), no. 4, 1051–1094. MR 1398816, DOI 10.1090/S0894-0347-96-00216-0
[21] A. Wilkie, A general theorem of the complement and some new o-minimal structures, preprint, 1996.
- [1]
- R. Benedetti and J.-J. Risler, Real algebraic and semi-algebraic sets, Hermann, 1990. MR 1070358
- [2]
- E. Bierstone and P. Milman, Semianalytic and subanalytic sets, IHES Publ. Math 67 (1988), 5-42. MR 0972342
- [3]
- J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry, Springer Verlag, 1998. MR 1659509
- [4]
- L. van den Dries, Remarks on Tarski's problem concerning
, in Logic Colloquium '82, G. Lolli, G. Longo and A. Marcja, eds., North-Holland, 1984, 97-121. MR 0762106
- [5]
- L. van den Dries, A generalization of the Tarski-Seidenberg theorem, and some nondefinability results, Bull. AMS 15 (1986), 189-193. MR 0854552
- [6]
- L. van den Dries, A. Macintyre and D. Marker, The elementary theory of restricted analytic fields with exponentiation, Ann. Math 140 (1994), 183-205. MR 1289495
- [7]
- L. van den Dries and C. Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), 497-540. MR 1404337
- [8]
- L. van den Dries and P. Speissegger, The real field with convergent generalized power series, Trans. AMS 350 (1998), 4377-4421. MR 1458313
- [9]
- L. van den Dries and P. Speissegger, The field of reals with multisummable series and the exponential function, Proc. London Math. Soc., to appear.
- [10]
- A. Grothendieck, Esquisse d'un Programme, in [17], 5-48. MR 1483107
- [11]
- A. Khovanskii, On a class of systems of transcendental equations, Sov. Math. Dokl. 2 (1980) 762-765. MR 0600749
- [12]
- J. Knight, A. Pillay and C. Steinhorn, Definable sets in ordered structures II, Trans. AMS 295 (1986) 593-605. MR 0833697
- [13]
- Y. Peterzil and S. Starchenko, A trichotomy theorem for o-minimal structures, Proc. London Math. Soc. 77 (1998), 481-523. MR 1643405
- [14]
- Y. Peterzil, A. Pillay and S. Starchenko, Simple algebraic and semialgebraic groups over real closed fields, Trans. AMS, to appear.
- [15]
- A. Pillay, On groups and fields definable in o-minimal structures, J. Pure. Appl. Algebra 53 (1988), 239-255. MR 0961362
- [16]
- A. Pillay and C. Steinhorn, Definable sets in ordered structures I, Trans. AMS 295 (1986), 565-592. MR 0833697
- [17]
- L. Schneps and P. Lochak, Geometric Galois Actions: I. Around Grothendieck's Esquisse d'un Programme, Cambridge Univ. Press, 1997. MR 1483106
- [18]
- P. Speissegger, The Pfaffian closure of an o-minimal structure, J. Reine Angew. Math. 508 (1999), 198-211. CMP 99:09
- [19]
- B. Tessier, Tame and stratified objects, in [17], 231-243.
- [20]
- A. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. AMS 9 (1996), 1051-1094. MR 1398816
- [21]
- A. Wilkie, A general theorem of the complement and some new o-minimal structures, preprint, 1996.
Review Information:
Reviewer:
David Marker
Affiliation:
University of Illinois at Chicago
Email:
marker@math.uic.edu
Journal:
Bull. Amer. Math. Soc.
37 (2000), 351-357
DOI:
https://doi.org/10.1090/S0273-0979-00-00866-1
Published electronically:
March 2, 2000
Review copyright:
© Copyright 2000
American Mathematical Society
