Celebratio Mathematica

L. P. D. van den Dries: Model theory of fields: Decidability and bounds for polynomial ideals. Ph.D. thesis, Rijksuniversiteit Utrecht, 1978. phdthesis

L. van den Dries and K. Schmidt: “Bounds in the theory of polynomial rings over fields: A nonstandard approach,” Invent. Math. 76 : 1 (February 1984), pp. 77–91. MR 739626 Zbl 0539.13011 article

L. van den Dries: “Remarks on Tarski’s problem concerning \( (\mathbb{R},+,\cdot\,,\operatorname{exp}) \),” pp. 97–121 in Logic colloquium ’82 (Florence, 23–28 August 1982). Edited by G. Lolli, G. Longo, and A. Marcja. Studies in Logic and the Foundations of Mathematics 112. North-Holland (Amsterdam), 1984. MR 762106 Zbl 0585.03006 incollection

Gabriele Lolli	Related
Giuseppe Longo	Related
Annalisa Marcja	Related

J. Denef and L. van den Dries: “\( p \)-adic and real subanalytic sets,” Ann. Math. (2) 128 : 1 (July 1988), pp. 79–138. MR 951508 Zbl 0693.14012 article

L. van den Dries, A. Macintyre, and D. Marker: “The elementary theory of restricted analytic fields with exponentiation,” Ann. Math. (2) 140 : 1 (July 1994), pp. 183–205. MR 1289495 Zbl 0837.12006 article

In [1996] Wilkie proved the remarkable result that the field of real numbers with exponentiation is model complete. When we combine this with Hovanskiĭ’s finiteness theorem [1980], it follows that the real exponential field is \( o \)-minimal. In \( o \)-minimal expansions of the real field the definable subsets of \( \mathbb{R}^n \) share many of the nice structural properties of semialgebraic sets. For example, definable subsets have only finitely many connected components, definable sets can be stratified and triangulated, and continuous definable maps are piecewise trivial (see [van den Dries 1998]).

In this paper we will prove a quantifier elimination result for the real field augmented by exponentiation and all restricted analytic functions, and use this result to obtain \( o \)-minimality. We were led to this while studying work of Ressayre [1993] and several of his ideas emerge here in simplified form. However, our treatment is formally independent of the results of [Wilkie 1996; Hovanskiĭ 1980; Ressayre 1993].

David E. Marker	Related
Angus Macintyre	Related

L. van den Dries, A. Macintyre, and D. Marker: “Logarithmic-exponential power series,” J. London Math. Soc. (2) 56 : 3 (1997), pp. 417–434. MR 1610431 Zbl 0924.12007 article

David E. Marker	Related
Angus Macintyre	Related

M. Aschenbrenner, L. van den Dries, and J. van der Hoeven: Asymptotic differential algebra and model theory of transseries. Annals of Mathematics Studies 195. Princeton University Press, 2017. MR 3585498 Zbl 06684722 book

Joris van der Hoeven	Related
Matthias Aschenbrenner	Related

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Lou van den Dries

Interview with Lou van den Dries

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