Celebratio Mathematica

L. van den Dries, D. Marker, and G. Martin: “Definable equivalence relations on algebraically closed fields,” J. Symbolic Logic 54 : 3 (September 1989), pp. 928–935. MR 1011180 Zbl 0689.03019 article

David E. Marker	Related
Gary Alan Martin	Related

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

L. van den Dries, A. Macintyre, and D. Marker: “Logarithmic-exponential series,” pp. 61–113 in Proceedings of the international conference “Analyse & Logique” (Mons, Belgium, 25–29 August 1997), published as Ann. Pure Appl. Logic 111 : 1–2. Issue edited by C. Finet and C. Michaux. July 2001. MR 1848569 Zbl 0998.12014 incollection

We extend the field of Laurent series over the reals in a canonical way to an ordered differential field of “logarithmic-exponential series” (LE-series), which is equipped with a well behaved exponentiation. We show that the LE-series with derivative 0 are exactly the real constants, and we invert operators to show that each LE-series has a formal integral. We give evidence for the conjecture that the field of LE-series is a universal domain for ordered differential algebra in Hardy fields. We define composition of LE-series and establish its basic properties, including the existence of compositional inverses. Various interesting subfields of the field of LE-series are also considered.

David E. Marker	Related
Christian Michaux	Related
Catherine Finet	Related
Angus Macintyre	Related

L. van den Dries and P. Speissegger: “o-minimal preparation theorems,” pp. 87–116 in Model theory and applications (Ravello, Italy, 27 May–1 June 2002). Edited by L. Bélair, Z. Chatzidakis, P. D’Aquino, D. Marker, M. Otero, F. Point, and A. Wilkie. Quaderni di Matematica 11. Aracne (Rome), 2002. To Angus Macintyre, on his 60th birthday. MR 2159715 Zbl 1081.03039 incollection

David E. Marker	Related
Zoé Maria Chatzidakis	Related
Paola D'Aquino	Related
Angus Macintyre	Related
Alex J. Wilkie	Related
Margarita Otero	Related
Françoise Point	Related
Patrick Speissegger	Related
Luc Bélair	Related

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