Hostname: page-component-7dc689bd49-bfm8c Total loading time: 0 Render date: 2023-03-21T02:42:49.470Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

Model Theory of Analytic Functions: Some Historical Comments

Published online by Cambridge University Press:  15 January 2014

Deirdre Haskell*
Department of Mathematics and Statistics, Mcmaster University, Hamilton, ON L8S 4K1, CanadaE-mail:


Model theorists have been studying analytic functions since the late 1970s. Highlights include the seminal work of Denef and van den Dries on the theory of the p-adics with restricted analytic functions, Wilkie's proof of o-minimality of the theory of the reals with the exponential function, and the formulation of Zilber's conjecture for the complex exponential. My goal in this talk is to survey these main developments and to reflect on today's open problems, in particular for theories of valued fields.

Research Article
Copyright © Association for Symbolic Logic 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)



[1] Aschenbrenner, M., Review of [45]. mr2102856, 2006.Google Scholar
[2] Ax, J., On the undecidability of power series fields, Proceedings of the American Mathematical Society, vol. 16 (1965), p. 846.Google Scholar
[3] Ax, J., On Schanuel's conjectures, Annals of Mathematics, vol. 93 (1971), no. 2, pp. 252268.CrossRefGoogle Scholar
[4] Ax, J. and Kochen, S., Diophantine problems over local fields. III. Decidable fields, Annals of Mathematics, vol. 83 (1966), no. 2, pp. 437456.CrossRefGoogle Scholar
[5] Berkovich, V., p-adic analytic spaces, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Documenta Mathematica, 1998, Extra Vol. II, pp. 141151.Google Scholar
[6] Bierstone, E. and Milman, P., Semianalytic and subanalytic sets, Institut des Hautes Études Scientifiques. Publications Mathématiques, vol. 67 (1988), p. 542.CrossRefGoogle Scholar
[7] Blum, L., Differentially closed fields: a model-theoretic tour, Contributions to algebra (collection of papers dedicated to Ellis Kolchin), Academic Press, New York, 1977, pp. 3761.Google Scholar
[8] Cohen, P., Decision procedures for real and p-adic fields, Communications in Pure and Applied Mathematics, vol. 22 (1969), p. 131151.CrossRefGoogle Scholar
[9] Denef, J. and van den Dries, L., p-adic and real subanalytic sets, Annals of Mathematics. Second Series, vol. 128 (1988), no. 1, pp. 79138.CrossRefGoogle Scholar
[10] Ersov, Ju., On elementary theories of local fields, Algebra i Logika Sem. 4, (1965), no. 2, pp. 530.Google Scholar
[11] GabrieLov, A., Projections of semianalytic sets, Akademija Nauk SSSR. Funkcional'nyi Analiz i ego Priloženija, vol. 2 (1968), no. 4, pp. 1830, Russian; also Functional Analysis and Its Applications , vol. 2 (1968), pp. 282–291 (English version).Google Scholar
[12] Hardt, R., Stratification of real analytic mappings and images, Inventiones Mathematicae, vol. 28 (1975), p. 193208.CrossRefGoogle Scholar
[13] Haskell, D., Hrushovski, E., and Macpherson, D., Definable sets in algebraically closed valued fields: elimination of imaginaries, Journal für die Reine und Angewandte Mathematik, vol. 597 (2006), p. 175236.Google Scholar
[14] Hironaka, H., Subanalytic sets, Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 453493.Google Scholar
[15] Hrushovski, E. and Loeser, F., Non-archimedean tame topology and stably dominated types, arXiv:1009.0252.Google Scholar
[16] Hrushovski, E. and Martin, B., Zeta functions from definable equivalence relations, arXiv:math/0701011.Google Scholar
[17] Knight, J., Pillay, A., and Stemhorn, C., Definable sets in ordered structures II, Transactions of the American Mathematical Society, vol. 295 (1986), no. 2, pp. 593605.CrossRefGoogle Scholar
[18] Koiran, P., The theory of Liouville functions, The Journal of Symbolic Logic, vol. 68 (2005), no. 2, pp. 353365.CrossRefGoogle Scholar
[19] Lang, S., Introduction to transcendental numbers, Addison-Wesley, Reading, Massachusetts, 1966.Google Scholar
[20] Lipshitz, L., Rigid subanalytic sets, American Journal of Mathematics, vol. 115 (1993), no. 1, pp. 77108.CrossRefGoogle Scholar
[21] Łojasiewicz, S., Ensembles semi-analytiques, IHES Lecture Notes, Bures-sur-Yvette, France, 1965.Google Scholar
[22] Macintyre, A., On definable subsets of p-adic fields, The Journal of Symbolic Logic, vol. 41 (1976), no. 3, pp. 605610.CrossRefGoogle Scholar
[23] Macintyre, A. and Wilkie, A., On the decidability of the real exponential field, Kreiseliana, A K Peters, Wellesley, MA, 1996, pp. 441467.Google Scholar
[24] Marker, D., Model theory and exponentiation, Notices of the American Mathematical Society, vol. 43 (1996), no. 7, pp. 753759.Google Scholar
[25] Marker, D., A remark on Zilber's pseudo-exponentiation, The Journal of Symbolic Logic, vol. 71 (2006), p. 791798.CrossRefGoogle Scholar
[26] Mellor, T., Imaginaries in real closed valued fields, Annals of Pure and Applied Logic, vol. 139 (2006), no. 1–3, pp. 230279.CrossRefGoogle Scholar
[27] Moosa, R., The model theory of compact complex spaces, Logic colloquium '01 (Baaz, M., Friedman, S., and Krajicek, J., editors), Lecture Notes in Logic, vol. 20, Association for Symbolic Logic, 2005.Google Scholar
[28] Moosa, R., Model theory and complex geometry, Notices of the American Mathematical Society, vol. 57 (2010), no. 2, pp. 230235.Google Scholar
[29] Peterzil, Y. and Starchenko, S., Expansions of algebraically closedfields in o-minimal structures, Selecta Mathematica. New Series, vol. 7 (2001), no. 3, pp. 409445.CrossRefGoogle Scholar
[30] Ritt, J., Differential algebra, American Mathematical Society Colloquium Publications, XXXIII, American Mathematical Society, 1950.CrossRefGoogle Scholar
[31] Robinson, A., On the concept of a differentially closed field, Bulletin of the Research Council of Israel Section F, (1959), pp. 113128.Google Scholar
[32] Robinson, A., On the real closure of a Hardy field, Theory of sets and topology (in honour of Felix Hausdorff, 1868–1942), VEB Deutscher Verlag der Wissenschaften, Berlin, 1972, pp. 427433.Google Scholar
[33] Robinson, R., Undecidable rings, Transactions of the American Mathematical Society, vol. 70 (1951), p. 137159.CrossRefGoogle Scholar
[34] Rolin, J.-P., Speissegger, P., and Wilkie, A., Quasianalytic Denjoy–Carleman classes and o-minimality, Journal of the American Mathematical Society, vol. 16 (2003), no. 4, pp. 751777.CrossRefGoogle Scholar
[35] Speissegger, P., The Pfaffian closure of an o-minimal structure, Journal für die Reine und Angewandte Mathematik, vol. 508 (1999), p. 189211.CrossRefGoogle Scholar
[36] Tarski, A., A decision methodfor elementary algebra and geometry, 2nd ed., University of California Press, Berkeley and Los Angeles, California, 1951; Tarski Collected Papers , Birkhauser, vol. 3, pp. 297–368, 1986.Google Scholar
[37] van den Dries, L., Remarks on Tarski's problem concerning (R, +, ×, exp), Logic colloquium '82 (Florence, 1982), Studies in Logic and the Foundations of Mathematics, 112, North-Holland, 1984, pp. 97121.CrossRefGoogle Scholar
[38] van den Dries, L., A generalization of the Tarski–Seidenberg theorem, and some nondefinability results, Bulletin of the American Mathematical Society. New Series, vol. 15 (1986), no. 2, pp. 189193.CrossRefGoogle Scholar
[39] van den Dries, L., Classical model theory of fields, Model theory, algebra and geometry, >MSRI 39 Cambridge University Press, 2000.Google Scholar
[40] Wilkie, A., Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, Journal of the American Mathematical Society, vol. 9 (1996), no. 4, pp. 10511094.CrossRefGoogle Scholar
[41] Wilkie, A., Model theory of analytic and smooth functions, Models and Computability, Proceedings of Logic Colloquium 1997, Cambridge University Press, 1997.Google Scholar
[42] Wilkie, A., A theorem of the complement and some new o-minimal structures, Selecta Mathematica. New Series, vol. 5 (1999), no. 4, pp. 397421.CrossRefGoogle Scholar
[43] Wilkie, A., Liouville functions, Logic colloquium 2000, Lecture Notes in Logic, 19, Association for Symbolic Logic, Urbana, IL, 2005, pp. 383391.Google Scholar
[44] Zilber, B., Model theory and algebraic geometry, Seminarberichte Nr 93-1, Proceedings of the 10th Easter Conference in Model Theory, April 12–17, 1993, Humboldt Universität zu Berlin, 1993,, pp. 202222.Google Scholar
[45] Zilber, B., Pseudo-exponentiation on algebraically closed fields of characteristic zero, Annals of Pure and Applied Logic, vol. 132 (2004), no. 1, pp. 6795.CrossRefGoogle Scholar