Celebratio Mathematica

L. van den Dries: “Artin–Schreier theory for commutative regular rings,” Ann. Math. Logic 12 : 2 (December 1977), pp. 113–150. MR 498094 Zbl 0376.13012 article

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

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

J. Becker, J. Denef, L. Lipshitz, and L. van den Dries: “Ultraproducts and approximation in local rings, I,” Invent. Math. 51 : 2 (June 1979), pp. 189–203. MR 528023 Zbl 0416.13004 article

Leonard Michael Lipshitz	Related
Joseph A. Becker	Related
Jan Denef	Related

L. van den Dries and P. Ribenboim: “Application de la théorie des modèles aux groupes de Galois de corps de fonctions” [Application of model theory to Galois groups of function fields], C. R. Acad. Sci. Paris Sér. A-B 288 : 17 (1979), pp. A789–A792. MR 535636 Zbl 0426.12004 article

L. van den Dries: “New decidable fields of algebraic numbers,” Proc. Am. Math. Soc. 77 : 2 (November 1979), pp. 251–256. MR 542093 Zbl 0396.12020 article

L. van den Dries: “Algorithms and bounds for polynomial rings,” pp. 147–157 in Logic colloquium ’78 (Mons, Belgium, 24 August–1 September 1978). Edited by M. Boffa, D. Dalen, and K. McAloon. Studies in Logic and the Foundations of Mathematics 97. North-Holland (Amsterdam and New York), 1979. MR 567669 Zbl 0461.13015 incollection

Maurice Boffa	Related
Kenneth Walter McAloon	Related
Dirk van Dalen	Related

L. P. D. van den Dries: “A linearly ordered ring whose theory admits elimination of quantifiers is a real closed field,” Proc. Am. Math. Soc. 79 : 1 (May 1980), pp. 97–100. MR 560592 Zbl 0397.06017 article

L. van den Dries: “Some model theory and number theory for models of weak systems of arithmetic,” pp. 346–362 in Model theory of algebra and arithmetic (Karpacz, Poland, 1–7 September 1979). Edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie. Lecture Notes in Mathematics 834. Springer, 1980. MR 606793 Zbl 0454.03034 incollection

Jędrzej Wierzejewski	Related
Leszek Pacholski	Related
Alex J. Wilkie	Related

G. Cherlin, L. van den Dries, and A. Macintyre: “Decidability and undecidability theorems for PAC-fields,” Bull. Am. Math. Soc. (N.S.) 4 : 1 (1981), pp. 101–104. MR 590820 Zbl 0466.12017 article

A pseudo-algebraically closed field (PAC-field for short) is a field \( K \) such that every absolutely irreducible variety defined over \( K \) has \( K \)-rational points. In [1968] Ax gave a decision method for the (elementary) theory of finite fields, the basis of which was his characterization of the infinite models of that theory as the perfect PAC-fields \( K \) with \[ G(K)\cong \hat{\mathbf{Z}} .\] (Here and in the following we use the notations \( G(K) \) for the absolute Galois group \[ \mathrm{Gal}(\tilde{K}\vert K) \] of \( K \), \( \tilde{K} = \) algebraic closure of \( K \), \( G = \) profinite completion of the discrete group \( G \).) In [1969] M. Jarden gave another natural source of PAC-fields: let \( e\in\mathbf{N} \); then for almost all \( e \)-tuples \[ (\sigma_1,\dots,\sigma_e)\in (\mathbf{Q})^e \] — in the sense of the Haar measure on \( G(\mathbf{Q}) \) — the fixed field \[ \mathrm{Fix}(\sigma_1,\dots,\sigma_e)\subset \tilde{\mathbf{Q}} \] is PAC, and \( e \)-free (where we call a field \( K \) \( e \)-free if \[ G(K)\cong \hat{F}_e ,\] \( F_e = \) free group on \( e \) generators). In 1975 Jarden and Kiehne classified \( e \)-free perfect PAC-fields up to elementary equivalence and derived the decidability of the theory of \( e \)-free perfect PAC-fields, cf. [1975]. The next step was an extension of these results to a certain class of PAC-fields with infinitely generated absolute Galois group, cf. [Jarden 1976]. We announce here solutions to the main questions provoked by this development.

Gregory L. Cherlin	Related
Angus Macintyre	Related

L. van den Dries: “Which curves over \( \mathbf{Z} \) have points with coordinates in a discrete ordered ring?,” Trans. Am. Math. Soc. 264 : 1 (March 1981), pp. 181–189. MR 597875 Zbl 0464.10047 article

A. Lubotzky and L. van den Dries: “Subgroups of free profinite groups and large subfields of \( \tilde{\mathbf{Q}} \),” Israel J. Math. 39 : 1–2 (March 1981), pp. 25–45. MR 617288 Zbl 0485.20021 article

L. van den Dries: “Quantifier elimination for linear formulas over ordered and valued fields,” Bull. Soc. Math. Belg. Sér. B 33 : 1 (1981), pp. 19–31. MR 620959 Zbl 0479.03018 article

L. van den Dries: “A specialization theorem for \( p \)-adic power series converging on the closed unit disc,” J. Algebra 73 : 2 (December 1981), pp. 613–623. MR 640053 Zbl 0511.12018 article

L. P. D. van den Dries, A. M. W. Glass, A. Macintyre, A. H. Mekler, and J. Poland: “Elementary equivalence and the commutator subgroup,” Glasgow Math. J. 23 : 2 (July 1982), pp. 115–117. MR 663136 Zbl 0504.03006 article

Alan Harvey Mekler	Related
Andrew Martin William Glass	Related
John C. Poland	Related
Angus Macintyre	Related

L. van den Dries: “A specialization theorem for analytic functions on compact sets,” Indag. Math. 85 : 4 (December 1982), pp. 391–396. MR 683526 Zbl 0526.30004 article

L. van den Dries: “Some applications of a model theoretic fact to (semi-)algebraic geometry,” Indag. Math. 85 : 4 (December 1982), pp. 397–401. MR 683527 Zbl 0538.14017 article

Logicians are familiar with Tarski’s quantifier elimination for the theories of algebraically closed and real closed fields. Sometimes this result is not precise enough: the problem may be that a certain set has to be shown definable, not just by a q.f. (= quantifier free) formula, but by apositive q.f. formula (i.e., by a conjunction of equations in the case of algebraic geometry, by a disjunction of systems of weak inequalities \( f_1\geq 0 \),…, \( f_k\geq 0 \) in the case of semi-algebraic geometry).

In this paper we indicate a simple model theoretic way to obtain representations by positive q.f. formulas: we prove a Lyndon-type theorem characterizing those formulas which are equivalent to a positive q.f. formula (relative to a given theory); in the case of interest one just applies the place extension theorem for (ordered) integral domains to verify the hypothesis of our model theoretic result.

A. Macintyre, K. McKenna, and L. van den Dries: “Elimination of quantifiers in algebraic structures,” Adv. Math. 47 : 1 (January 1983), pp. 74–87. MR 689765 Zbl 0531.03016 article

Kenneth Gerard McKenna	Related
Angus Macintyre	Related

E. Connell and L. van den Dries: “Injective polynomial maps and the Jacobian conjecture,” J. Pure Appl. Algebra 28 : 3 (June 1983), pp. 235–239. MR 701351 Zbl 0513.13007 article

L. van den Dries: Reducing to prime characteristic by means of Artin approximation and constructible properties, and applied to Hochster algebras. Communications of the Mathematical Institute, Rijksuniversiteit Utrecht 16. Rijksuniversiteit Utrecht, 1983. With a preface by Jan R. Strooker. MR 710483 Zbl 0513.13010 book

L. van den Dries: “Analytic Hardy fields and exponential curves in the real plane,” Am. J. Math. 106 : 1 (February 1984), pp. 149–167. MR 729758 Zbl 0597.26002 article

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: “Algebraic theories with definable Skolem functions,” J. Symbolic Logic 49 : 2 (June 1984), pp. 625–629. MR 745390 Zbl 0596.03032 article

L. van den Dries: “Exponential rings, exponential polynomials and exponential functions,” Pacific J. Math. 113 : 1 (1984), pp. 51–66. MR 745594 Zbl 0603.13019 article

L. van den Dries and P. Ribenboim: “The absolute Galois group of a rational function field in characteristic zero is a semidirect product,” Canad. Math. Bull. 27 : 3 (September 1984), pp. 313–315. MR 749638 Zbl 0548.12013 article

L. van den Dries and A. J. Wilkie: “Gromov’s theorem on groups of polynomial growth and elementary logic,” J. Algebra 89 : 2 (August 1984), pp. 349–374. MR 751150 Zbl 0552.20017 article

L. van den Dries and H. Levitz: “On Skolem’s exponential functions below \( 2^{2^X} \),” Trans. Am. Math. Soc. 286 : 1 (November 1984), pp. 339–349. MR 756043 Zbl 0556.03036 article

A result of Ehrenfeucht implies that the smallest class of number-theoretic functions \[ f:\mathbf{N}\to\mathbf{N} \] containing the constants \( 0,\,1,\,2,\dots \), the identity function \( X \), and closed under addition, multiplication and \( f\to f^X \), is well-ordered by the relation of eventual dominance. We show that its order type is \( \omega^{\omega^{\omega}} \), and that for any two nonzero functions \( f,\,g \) in the class the quotient \( f(n)/g(n) \) tends to a limit in \[ E^+\cup\{0,\infty\} \] as \( n\to\infty \), where \( E^+ \) is the smallest set of positive real numbers containing 1 and closed under addition, multiplication and under the operations \( x\to x^{-1} \), \( x\to e^x \).

A. J. Wilkie and L. van den Dries: “An effective bound for groups of linear growth,” Arch. Math. (Basel) 42 : 5 (May 1984), pp. 391–396. MR 756689 Zbl 0567.20016 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

L. van den Dries and R. L. Smith: “Decidable regularly closed fields of algebraic numbers,” J. Symbolic Logic 50 : 2 (June 1985), pp. 468–475. MR 793127 Zbl 0574.12023 article

A field \( K \) is regularly closed if every absolutely irreducible affine variety defined over \( K \) has \( K \)-rational points. This notion was first isolated by Ax [1968] in his work on the elementary theory of finite fields. Later Jarden [1976] and Jarden and Kiehne [1975] extended this in different directions. One of the primary results in this area is that the elementary properties of a regularly closed field \( K \) with a free Galois group (on either finitely or countably many generators) are determined by the set of integer polynomials in one indeterminate with a zero in \( K \). The method of proof employed in [Jarden 1972, 1976; Jarden and Kiehne 1975] is unusual for algebra since it is a measure-theoretic argument. In this brief summary we have not made any attempt at completeness. We refer the reader to the recent paper of Cherlin, van den Dries, and Macintyre [1980] and to the forthcoming book by Fried and Jarden [1986] for a more thorough discussion of the latest results.

L. van den Dries: “The field of reals with a predicate for the powers of two,” Manusc. Math. 54 : 1–2 (March 1985), pp. 187–195. MR 808687 Zbl 0631.03020 article

L. van den Dries: “A completeness theorem for trigonometric identities and various results on exponential functions,” Proc. Am. Math. Soc. 96 : 2 (February 1986), pp. 345–352. MR 818470 Zbl 0619.03012 article

L. van den Dries: “A generalization of the Tarski–Seidenberg theorem, and some nondefinability results,” Bull. Am. Math. Soc. (N.S.) 15 : 2 (1986), pp. 189–193. MR 854552 Zbl 0612.03008 article

L. van den Dries: “Tarski’s problem and Pfaffian functions,” pp. 59–90 in Logic colloquium ’84 (Manchester, UK, 15–24 July 1984). Edited by J. B. Paris, A. J. Wilkie, and G. M. Wilmers. Studies in Logic and the Foundations of Mathematics 120. North-Holland (Amsterdam), 1986. MR 861419 Zbl 0616.03018 incollection

George M. Wilmers	Related
Jeffrey Bruce Paris	Related
Alex J. Wilkie	Related

L. van den Dries and P. Ribenboim: “An application of Tarski’s principle to absolute Galois groups of function fields,” pp. 131–148 in Algebra and order (Luminy, France, 1984). Edited by S. Wolfenstein. Research and Exposition in Mathematics 14. Heldermann (Berlin), 1986. MR 891455 Zbl 0645.12010 incollection

The subject matter of this paper is the Galois theory of function fields, the methods involve the transfer principle of Lefschetz and similar ones.

This article has been circulating in preprint form since 1976 (Queen’s Mathematics Preprint No. 1976-5) and some of the results have in the meantime been published by the authors [1979] (with other proofs) and by Schuppar [1980]. Yet, the recent developments of real algebraic geometry and the kind encouragement of M. Knebusch have prompted us to present it at the occasion of the First International Symposium on Ordered Algebraic Structures, held in Luminy, June 1984. Indeed, we note especially our semi-algebraic definition of the fundamental group, and the semi-algebraic path-lifting and monodromy theorems, which anticipated but did not influence the more general development of semi-algebraic topology by Delfs and Knebusch.

Paulo Ribenboim	Related
Samuel Wolfenstein	Related

L. van den Dries and P. Ribenboim: “An application of Tarski’s principle to absolute Galois groups of function fields,” Ann. Pure Appl. Logic 33 : 1 (1987), pp. 83–107. MR 870687 Zbl 0645.12009 article

The subject matter of this paper is the Galois theory of function fields, the methods involve the transfer principle of Lefschetz and similar ones.

This article has been circulating in preprint form since 1976 (Queen’s Mathematics Preprint No. 1976-5) and some of the results have in the meantime been published by the authors [1979] (with other proofs) and by Schuppar [1980]. Yet, the recent developments of real algebraic geometry and the kind encouragement of M. Knebusch have prompted us to present it at the occasion of the First International Symposium on Ordered Algebraic Structures, held in Luminy, June 1984. Indeed, we note especially our semi-algebraic definition of the fundamental group, and the semi-algebraic path-lifting and monodromy theorems, which anticipated but did not influence the more general development of semi-algebraic topology by Delfs and Knebusch.

L. van den Dries: “Alfred Tarski’s elimination theory for real closed fields,” J. Symbolic Logic 53 : 1 (March 1988), pp. 7–19. MR 929371 Zbl 0651.03001 article

Tarski made a fundamental contribution to our understanding of \( \mathbf{R} \), perhaps mathematics’ most basic structure. His theorem is the following.

To any formula \[ \phi(X_1,\dots,X_m) \] in the vocabulary \( \{0,1,+,\cdot\,, < \} \) one can effectively associate two objects:

a quantifier free formula \( \overline{\phi}(X_1,\dots,X_m) \) in the same vocabulary, and
a proof of the equivalence \( \phi \leftrightarrow \overline{\phi} \) that uses only the axioms for real closed fields.

(Reminder: real closed fields are ordered fields with the intermediate value property for polynomials.)

Everything in (i) has turned out to be crucial: that arbitrary formulas are considered rather than just sentences, that the equivalence \( \phi \leftrightarrow \overline{\phi} \) holds in all real closed fields rather than only in \( \mathbb{R} \); even the effectiveness of the passage from \( \phi \) to \( \overline{\phi} \) has found good theoretical uses besides firing the imagination.

We begin this survey with some history in §1. In §2 we discuss three other influential proofs of Tarski’s theorem, and in §3 we consider some of the remarkable and totally unforeseen ways in which Tarski’s theorem functions nowadays in mathematics, logic and computer science.

L. van den Dries: “Elimination theory for the ring of algebraic integers,” J. Reine Angew. Math. 1988 : 388 (1988), pp. 189–205. MR 944190 Zbl 0659.12021 article

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: “On the elementary theory of restricted elementary functions,” J. Symbolic Logic 53 : 3 (September 1988), pp. 796–808. MR 960999 Zbl 0698.03023 article

L. Bélair, L. van den Dries, and A. Macintyre: “Elementary equivalence and codimension in \( p \)-adic fields,” Manuscripta Math. 62 : 2 (June 1988), pp. 219–225. MR 963007 Zbl 0665.12027 article

Luc Bélair	Related
Angus Macintyre	Related

P. Scowcroft and L. van den Dries: “On the structure of semialgebraic sets over \( p \)-adic fields,” J. Symbolic Logic 53 : 4 (December 1988), pp. 1138–1164. MR 973105 Zbl 0692.14014 article

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: “Convergent series expansions of a new type for exponential functions,” Math. Nachr. 142 (1989), pp. 7–18. MR 1017368 Zbl 0709.13011 article

L. van den Dries: “Dimension of definable sets, algebraic boundedness and Henselian fields.” Edited by J. T. Baldwin and A. Marcja. Ann. Pure Appl. Logic 45 : 2 (1989), pp. 189–209. MR 1044124 Zbl 0704.03017 article

John Theodore Baldwin	Related
Annalisa Marcja	Related

L. P. D. van den Dries: “Weil’s group chunk theorem: A topological setting,” Ill. J. Math. 34 : 1 (1990), pp. 127–139. MR 1031890 Zbl 0764.22001 article

A. Weil showed that a birational group law that is only partially defined can be extended to an algebraic group [1948,1955]. We show below (§1) that a similar construction can be carried out in a topological setting, where the topology is not necessarily the Zariski topology. In §2 we enrich our topological spaces with sheaves and prove a version of Weil’s theorem for this “structured” setting. We then derive Weil’s original theorem as well as two variations, coveting the cases of “quasi-algebraic” group chunks and “differentially algebraic” group chunks (§3). We note that our theorem, though quite general, does not include the scheme theoretic version of Weil’s theorem given in [Artin 1970].

K. McKenna and L. van den Dries: “Surjective polynomial maps, and a remark on the Jacobian problem,” Manuscripta Math. 67 : 1 (December 1990), pp. 1–15. MR 1037991 Zbl 0714.14013 article

L. van den Dries and A. Macintyre: “The logic of Rumely’s local-global principle,” J. Reine Angew. Math. 1990 : 407 (1990), pp. 33–56. MR 1048527 Zbl 0703.13021 article

L. van den Dries: “A remark on Ax’s theorem on solvability modulo primes,” Math. Z. 208 : 1 (December 1991), pp. 65–70. MR 1125733 Zbl 0744.11064 article

Z. Chatzidakis, L. van den Dries, and A. Macintyre: “Definable sets over finite fields,” J. Reine Angew. Math. 1992 : 427 (May 1992), pp. 107–135. MR 1162433 Zbl 0759.11045 article

Zoé Maria Chatzidakis	Related
Angus Macintyre	Related

L. van den Dries and J. Holly: “Quantifier elimination for modules with scalar variables,” Ann. Pure Appl. Logic 57 : 2 (May 1992), pp. 161–179. MR 1166465 Zbl 0772.03017 article

L. van den Dries: “Analytic Ax–Kochen–Ersov theorems,” pp. 379–398 in Proceedings of the international conference on algebra (Novosibirsk, USSR, 21–26 August 1989), Part 3. Edited by L. A. Bokut, Yu. L. Ershov, and A. I. Kostrikin. Contemporary Mathematics 131. American Mathematical Society (Providence, RI), 1992. Proceedings dedicated to the memory of A. I. Malcev. MR 1175894 Zbl 0835.03004 incollection

Yuriĭ Leonidovich Ershov	Related
Anatoliĭ Ivanovic Maltsev	Related
Leonid A. Bokut	Related
Aleksei Ivanovich Kostrikin	Related

C. W. Henson, L. A. Rubel, L. van den Dries, and M. F. Singer: “On the integer zeros of exponential polynomials,” Complex Variables Theory Appl. 23 : 3–4 (1993), pp. 201–211. MR 1269635 Zbl 0790.30003 article

C. Ward Henson	Related
Michael F. Singer	Related
Lee Albert Rubel	Related

L. van den Dries and C. Miller: “On the real exponential field with restricted analytic functions,” Israel J. Math. 85 : 1–3 (February 1994), pp. 19–56. A correction to this article was published in Israel J. Math. 92:1–3 (1995). MR 1264338 Zbl 0823.03017 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 and C. Miller: “Extending Tamm’s theorem,” Ann. Inst. Fourier (Grenoble) 44 : 5 (1994), pp. 1367–1395. MR 1313788 Zbl 0816.32004 article

L. van den Dries and A. H. Lewenberg: “\( T \)-convexity and tame extensions,” J. Symbolic Logic 60 : 1 (March 1995), pp. 74–102. MR 1324502 Zbl 0856.03028 article

Let \( T \) be a complete o-minimal extension of the theory of real closed fields. We characterize the convex hulls of elementary substructures of models of \( T \) and show that the residue field of such a convex hull has a natural expansion to a model of \( T \). We give a quantifier elimination relative to \( T \) for the theory of pairs \( (\mathscr{R},V) \) where \( \mathscr{R} \models T \) and \( V \neq \mathscr{R} \) is the convex hull of an elementary substructure of \( \mathscr{R} \). We deduce that the theory of such pairs is complete and weakly o-minimal. We also give a quantifier elimination relative to \( T \) for the theory of pairs with \( \mathscr{R} \) a model of \( T \) and a proper elementary substructure that is Dedekind complete in \( \mathscr{R} \). We deduce that the theory of such “tame” pairs is complete.

L. van den Dries: “Parametrizing the solutions of an analytic differential equation,” Ill. J. Math. 39 : 3 (1995), pp. 450–462. Dedicated to the memory of Lee A. Rubel (1928–1995). MR 1339837 Zbl 0845.34010 article

One of Rubel’s many research problems about algebraic differential equations (cf. [Rubel 1983, problem 21; Rubel 1992, problem 28]) is as follows:

Given a sequence \( (z_n) \) of distinct complex numbers tending to infinity, and any sequence \( (w_n) \) of complex numbers, does there exist a differentially algebraic entire function \( f \) such that \( f(z_n) = w_n \) , for all \( n\in \mathbf{N} \) ?

It is classical that the answer is “yes” if the “differentially algebraic” requirement is dropped. Below, we show the answer is “no” in our case:

Given any such sequence \( (z_n) \) , the set of sequences \( (w_n)\in \mathbf{C^N} \) for which there is a differentially algebraic entire function \( f \) with \( f(z_n) = w_n \) for all \( n \) , is meagre in the sequence space \( \mathbf{C^N} \) equipped with the product topology. (Rubel had already shown in [Rubel 1992] that if one prescribes not only the values of \( f(z_n) \) but also those of \( f^{(j)}(z_n) \) for \( 1\leq j\leq n \), then the interpolation problem is in general not solvable.)

The goal of this paper is to reproduce Malgrange’s local parametrization of the solutions of an analytic differential equation, which is perhaps not as widely known as it should be, and to indicate its role in answering questions of this sort.

L. van den Dries and C. Miller: “Correction to: ‘On the real exponential field with restricted analytic functions’,” Israel J. Math. 92 : 1–3 (February 1995), pp. 427. Correction to an article published in Israel J. Math. 85:1–3 (1994). MR 1357768 Zbl 0973.03514 article

L. van den Dries and C. Miller: “Geometric categories and o-minimal structures,” Duke Math. J. 84 : 2 (August 1996), pp. 497–540. MR 1404337 Zbl 0889.03025 article

L. van den Dries: “o-minimal structures on the field of real numbers,” Jahresber. Deutsch. Math.-Verein. 98 : 3 (1996), pp. 165–171. MR 1421024 Zbl 0854.03036 article

L. van den Dries: “o-minimal structures,” pp. 137–185 in Logic: From foundations to applications (Keele, UK, 20–29 July 1993). Edited by W. Hodges, M. Hyland, C. Steinhorn, and J. Truss. Oxford Science Publications. Oxford University Press (New York), 1996. European Logic Colloquium. MR 1428004 Zbl 0861.03028 incollection

Wilfrid Augustine Hodges	Related
J. Martin E. Hyland	Related
John Kenneth Truss	Related
Charles Ira Steinhorn	Related

L. van den Dries: “\( T \)-convexity and tame extensions, II,” J. Symbolic Logic 62 : 1 (March 1997), pp. 14–34. A correction to this article was published in J. Symbolic Logic 63:4 (1998). MR 1450511 Zbl 0922.03055 article

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 and P. Speissegger: “The real field with convergent generalized power series,” Trans. Am. Math. Soc. 350 : 11 (1998), pp. 4377–4421. MR 1458313 Zbl 0905.03022 article

L. van den Dries: “Dense pairs of o-minimal structures,” Fund. Math. 157 : 1 (1998), pp. 61–78. MR 1623615 Zbl 0906.03036 article

L. van den Dries: Tame topology and o-minimal structures. London Mathematical Society Lecture Note Series 248. Cambridge University Press, 1998. MR 1633348 Zbl 0953.03045 book

L. van den Dries: “Correction to: ‘\( T \)-convexity and tame extensions, II’,” J. Symbolic Logic 63 : 4 (1998), pp. 1597. Correction to an article published in J. Symbolic Logic 62:1 (1997). MR 1665787 Zbl 0927.03068 article

L. van den Dries: “On the elementary theory of rings of Witt vectors with a multiplicative set of representatives for the residue field,” Manuscripta Math. 98 : 2 (February 1999), pp. 133–137. MR 1667599 Zbl 0923.03053 article

L. van den Dries, D. Haskell, and D. Macpherson: “One-dimensional \( p \)-adic subanalytic sets,” J. London Math. Soc. (2) 59 : 1 (1999), pp. 1–20. MR 1688485 Zbl 0932.03038 article

Deirdre Haskell	Related
H. Dugald Macpherson	Related

L. van den Dries: “o-minimal structures and real analytic geometry,” pp. 105–152 in Current developments in mathematics, 1998 (Cambridge, MA, 1998). Edited by B. Mazur, W. Schmid, S. T. Yau, D. Jerison, I. M. Singer, and D. Stroock. International Press (Somerville, MA), 1999. MR 1772324 Zbl 0980.03043 incollection

Daniel W. Stroock	Related
Barry C. Mazur	Related
Wilfried Schmid	Related
Shing-Tung Yau	Related
Isadore M. Singer	Related	Volume
David Saul Jerison	Related

M. Aschenbrenner and L. van den Dries: “Closed asymptotic couples,” J. Algebra 225 : 1 (March 2000), pp. 309–358. MR 1743664 Zbl 0974.12015 article

The derivation of a Hardy field induces on its value group a certain function \( \Psi \) If a Hardy field extends the real field and is closed under powers, then its value group is also a vector space over \( \mathbb{R} \). Such “ordered vector spaces with \( \Psi \)-function” are called \( H \)-couples. We define closed \( H \)-couples and show that every \( H \)-couple can be embedded into a closed one. The key fact is that closed \( H \)-couples have an elimination theory: solvability of an arbitrary system of equations and inequalities (built up from vector space operations, the function \( \Psi \), parameters, and the unknowns to be solved for) is equivalent to an effective condition on the parameters of the system. The \( H \)-couple of a maximal Hardy field is closed, and this is also the case for the \( H \)-couple of the field of logarithmic-exponential series over \( \mathbb{R} \). We analyze in detail finitely generated extensions of a given \( H \)-couple.

L. van den Dries: “Classical model theory of fields,” pp. 37–52 in Model theory, algebra, and geometry. Edited by D. Haskell, A. Pillay, and C. Steinhorn. MSRI Publications 39. Cambridge University Press, 2000. MR 1773701 Zbl 0986.03034 incollection

We begin with some thoughts on how model theory relates to other parts of mathematics, and on the indirect role of Gödel’s incompleteness theorem in this connection. With this in mind we consider in Section 2 the fields of real and \( p \)-adic numbers and show how these algebraic objects are understood model-theoretically: theorems of Tarski, Kochen, and Macintyre. This leads naturally to a discussion of the famous work by Ax, Kochen and Ershov in the mid sixties on henselian fields and its number theoretic implications.

In Section 3 we add analytic structure to the real and \( p \)-adic fields, and indicate how results such as the Weierstrass preparation theorem can be used to extend much of Section 2 to this setting. Here we make contact with the theory of subanalytic sets developed by analytic geometers in the real case.

In Section 4 we focus on o-minimal expansions of the real field that are not subanalytic, such as the real exponential field (Wilkie’s theorem). We indicate in a diagram the main known o-minimal expansions of the real field. We also provide a translation into the coordinate-free language of manifolds via “analytic-geometric categories”. (This has been found useful by geometers.)

Deirdre Haskell	Related
Anand Pillay	Related
Charles Ira Steinhorn	Related

L. van den Dries and P. Speissegger: “The field of reals with multisummable series and the exponential function,” Proc. London Math. Soc. (3) 81 : 3 (2000), pp. 513–565. MR 1781147 Zbl 1062.03029 article

L. van den Dries: “An intermediate value property for first-order differential polynomials,” pp. 95–105 in Connections between model theory and algebraic and analytic geometry. Edited by A. Macintyre. Quaderni di Matematica 6. Aracne (Rome), 2000. MR 1930683 Zbl 0994.26005 incollection

L. van den Dries and P. Ehrlich: “Fields of surreal numbers and exponentiation,” Fund. Math. 167 : 2 (2001), pp. 173–188. MR 1816044 Zbl 0974.03035 article

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. Ehrlich: “Erratum to: ‘Fields of surreal numbers and exponentiation’,” Fund. Math. 168 : 3 (2001), pp. 295–297. MR 1853411 article

L. van den Dries and F.-V. Kuhlmann: “Images of additive polynomials in \( \mathbb{F}_q(\!(t)\!) \) have the optimal approximation property,” Canad. Math. Bull. 45 : 1 (2002), pp. 71–79. MR 1884135 Zbl 1009.12008 article

M. Aschenbrenner and L. van den Dries: “\( H \)-fields and their Liouville extensions,” Math. Z. 242 : 3 (2002), pp. 543–588. MR 1985465 Zbl 1066.12002 article

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

L. van den Dries and A. J. Wilkie: “The laws of integer divisibility, and solution sets of linear divisibility conditions,” J. Symb. Logic 68 : 2 (June 2003), pp. 503–526. MR 1976588 Zbl 1056.03020 article

L. van den Dries: “Generating the greatest common divisor, and limitations of primitive recursive algorithms,” Found. Comput. Math. 3 : 3 (August 2003), pp. 297–324. MR 1989479 Zbl 1019.03027 article

L. van den Dries and Y. N. Moschovakis: “Is the Euclidean algorithm optimal among its peers?,” Bull. Symbolic Logic 10 : 3 (September 2004), pp. 390–418. MR 2083290 Zbl 1095.03025 article

M. Aschenbrenner and L. van den Dries: “Liouville closed \( H \)-fields,” J. Pure Appl. Algebra 197 : 1–3 (May 2005), pp. 83–139. MR 2123981 Zbl 1134.12004 article

\( H \)-fields are fields with an ordering and a derivation subject to some compatibilities. (Hardy fields extending \( \mathbb{R} \) and fields of transseries over \( \mathbb{R} \) are \( H \)-fields.) We prove basic facts about the location of zeros of differential polynomials in Liouville closed \( H \)-fields, and study various constructions in the category of \( H \)-fields: closure under powers, constant field extension, completion, and building \( H \)-fields with prescribed constant field and \( H \)-couple. We indicate difficulties in obtaining a good model theory of \( H \)-fields, including an undecidability result. We finish with open questions that motivate our work.

M. Aschenbrenner and L. van den Dries: “Asymptotic differential algebra,” pp. 49–85 in Analyzable functions and applications (Edinburgh, UK, 17–21 June 2002). Edited by O. Costin, M. D. Kruskal, and A. Macintyre. Contemporary Mathematics 373. American Mathematical Society (Providence, RI), 2005. MR 2130825 Zbl 1087.12002 incollection

Ovidiu Costin	Related
Matthias Aschenbrenner	Related
Martin David Kruskal	Related
Angus Macintyre	Related

M. Aschenbrenner, L. van den Dries, and J. van der Hoeven: “Differentially algebraic gaps,” Selecta Math. (N.S.) 11 : 2 (December 2005), pp. 247–280. MR 2183848 Zbl 1151.12002 article

\( H \)-fields are ordered differential fields that capture some basic properties of Hardy fields and fields of transseries. Each \( H \)-field is equipped with a convex valuation, and solving first-order linear differential equations in \( H \)-field extensions is strongly affected by the presence of a “gap” in the value group. We construct a real closed \( H \)-field that solves every first-order linear differential equation, and that has a differentially algebraic \( H \)-field extension with a gap. This answers a question raised in [Aschenbrenner and van den Dries 2002]. The key is a combinatorial fact about the support of transseries obtained from iterated logarithms by algebraic operations, integration, and exponentiation.

Joris van der Hoeven	Related
Matthias Aschenbrenner	Related

L. van den Dries and A. Günaydın: “The fields of real and complex numbers with a small multiplicative group,” Proc. London Math. Soc. (3) 93 : 1 (July 2006), pp. 43–81. MR 2235481 Zbl 1101.03028 article

L. van den Dries: “On explicit definability in arithmetic,” pp. 65–86 in Logic in Tehran (Tehran, 18–22 October 2003). Edited by A. Enayat, I. Kalantari, and M. Moniri. Lecture Notes in Logic 26. Association of Symbolic Logic (La Jolla, CA), 2006. MR 2262314 Zbl 1107.03042 incollection

Mojtaba Moniri	Related
Iraj Kalantari	Related
Ali Enayat	Related

L. van den Dries: “Isomorphism of complete local Noetherian rings and strong approximation,” Proc. Am. Math. Soc. 136 : 10 (2008), pp. 3435–3448. MR 2415027 Zbl 1154.13006 article

L. van den Dries and Y. N. Moschovakis: “Arithmetic complexity,” ACM Trans. Comput. Log. 10 : 1 (2009). Article 2, 49 pp. MR 2537732 Zbl 1367.68116 article

L. van den Dries and S. Gao: “A Polish group without Lie sums,” Abh. Math. Semin. Univ. Hambg. 79 : 1 (June 2009), pp. 135–147. MR 2541347 Zbl 1181.22006 article

L. van den Dries and I. Goldbring: “Locally compact contractive local groups,” J. Lie Theory 19 : 4 (2009), pp. 685–695. MR 2598999 Zbl 1222.22004 ArXiv 0909.4565 article

L. van den Dries and A. Günaydın: “Mann pairs,” Trans. Am. Math. Soc. 362 : 5 (2010), pp. 2393–2414. MR 2584604 Zbl 1192.03011 article

Mann proved in the 1960s that for any \( n \geq 1 \) there is a finite set \( E \) of \( n \)-tuples \( (\eta_1,\dots \), \( \eta_n) \) of complex roots of unity with the following property: if \( a_1,\dots \), \( a_n \) are any rational numbers and \( \zeta_1,\dots \), \( \zeta_n \) are any complex roots of unity such that \[ \sum_{i=1}^n a_i\zeta_i = 1 \quad\text{and}\quad \sum_{i\in I} a_i \zeta_i\neq 0 \] for all nonempty \( I\subseteq \{1,\dots,n\} \), then \[ (\zeta_1,\dots,\zeta_n)\in E .\] Taking an arbitrary field \( \mathbf{k} \) instead of \( \mathbb{Q} \) and any multiplicative group in an extension field of \( \mathbf{k} \) instead of the group of roots of unity, this property defines what we call a Mann pair \( (\mathbf{k},\Gamma) \). We show that Mann pairs are robust in certain ways, construct various kinds of Mann pairs, and characterize them model-theoretically.

L. van den Dries and J. Maříková: “Triangulation in o-minimal fields with standard part map,” Fund. Math. 209 : 2 (2010), pp. 133–155. MR 2660560 Zbl 1221.03030 ArXiv 0901.2339 article

L. van den Dries and V. C. Lopes: “Division rings whose vector spaces are pseudofinite,” J. Symbolic Logic 75 : 3 (September 2010), pp. 1087–1090. MR 2723784 Zbl 1201.03021 article

L. van den Dries and I. Goldbring: “Globalizing locally compact local groups,” J. Lie Theory 20 : 3 (2010), pp. 519–524. An erratum to this article was published in J. Lie Theory 22:2 (2012). MR 2743102 Zbl 1203.22006 ArXiv 1003.0963 article

S. Azgin and L. van den Dries: “Elementary theory of valued fields with a valuation-preserving automorphism,” J. Inst. Math. Jussieu 10 : 1 (January 2011), pp. 1–35. MR 2749570 Zbl 1235.03068 article

L. van den Dries and V. C. Lopes: “Invariant measures on groups satisfying various chain conditions,” J. Symbolic Logic 76 : 1 (March 2011), pp. 209–226. MR 2791344 Zbl 1220.03024 article

L. van den Dries and A. Günaydın: “Erratum to ‘Mann pairs’,” Trans. Am. Math. Soc. 363 : 9 (2011), pp. 5057. Erratum to an article published in Trans. Am. Math. Soc. 362:5 (2010). MR 2806701 Zbl 1227.03049 article

L. van den Dries and A. Günaydın: “Definable sets in Mann pairs,” Comm. Algebra 39 : 8 (2011), pp. 2752–2763. MR 2834128 Zbl 1243.03054 article

Consider structures \( (\Omega,\mathbf{k},\Gamma) \) where \( \Omega \) is an algebraically closed field of characteristic zero, \( \mathbf{k} \) is a subfield, and \( \Gamma \) is a subgroup of the multiplicative group of \( \Omega \). Certain pairs \( (\mathbf{k},\Gamma) \) have been singled out as Mann pairs in [van den Dries and Günaydın 2010]. We give new examples of such Mann pairs, we axiomatize for each Mann pair \( (\mathbf{k},\Gamma) \) the first-order theory of \( (\Omega,\mathbf{k},\Gamma) \) in a cleaner way than in [van den Dries and Günaydın 2010], and, as the main result of the article, we characterize the subsets of \( \Omega^n \) that are definable in \( (\Omega,\mathbf{k},\Gamma) \).

L. van den Dries and I. Goldbring: “Erratum to ‘Globalizing locally compact local groups’,” J. Lie Theory 22 : 2 (2012), pp. 489–490. Erratum to an article published in J. Lie Theory 20:3 (2010). MR 2976929 Zbl 1242.22005 article

M. Aschenbrenner, L. van den Dries, and J. van der Hoeven: “Toward a model theory for transseries,” Notre Dame J. Form. Log. 54 : 3–4 (2013), pp. 279–310. For Anand Pillay, on his 60th birthday. MR 3091660 Zbl 1314.03037 article

Anand Pillay	Related
Joris van der Hoeven	Related
Matthias Aschenbrenner	Related

L. van den Dries, J. Koenigsmann, H. D. Macpherson, A. Pillay, C. Toffalori, and A. J. Wilkie: Model theory in algebra, analysis and arithmetic (Cetraro, Italy, 2–16 July 2012). Edited by H. D. Macpherson and C. Toffalori. Lecture Notes in Mathematics 2111. Springer (Heidelberg), 2014. MR 3013956 Zbl 1326.03045 book

Alex J. Wilkie	Related
Anand Pillay	Related
Jochen Koenigsmann	Related
Carlo Toffalori	Related
H. Dugald Macpherson	Related

L. van den Dries: “Truncation in Hahn fields,” pp. 579–595 in Valuation theory in interaction (Segovia and El Escorial, Spain, 18–29 July 2011). Edited by A. Campillo, F.-V. Kuhlmann, and B. Teissier. EMS Series of Congress Reports 10. European Mathematical Society (Zürich), 2014. MR 3329048 Zbl 1361.12004 incollection

Antonio Campillo López	Related
Bernard Teissier	Related
Franz-Viktor Kuhlmann	Related

L. van den Dries: “Lectures on the model theory of valued fields,” pp. 55–157 in Model theory in algebra, analysis and arithmetic (Cetraro, Italy, 2–16 July 2012). Edited by H. D. Macpherson and C. Toffalori. Lecture Notes in Mathematics 2111. Springer (Berlin), 2014. MR 3330198 Zbl 1347.03074 incollection

H. Dugald Macpherson	Related
Carlo Toffalori	Related

L. van den Dries: “Approximate groups [according to Hrushovski and Breuillard, Green, Tao],” pp. 79–113 in Séminaire Bourbaki: 2013/2014, exposés 1074–1088. Astérisque 367–368. Société Mathématique De France (Paris), 2015. Exposé no. 1077. MR 3363589 Zbl 1358.11024 incollection

Terence Chi-Shen Tao	Related
Ehud Hrushovski	Related
Ben Joseph Green	Related
Emmanuel François Breuillard	Related

L. van den Dries and I. Goldbring: “Hilbert’s 5th problem,” Enseign. Math. 61 : 1–2 (2015), pp. 3–43. MR 3449281 Zbl 1335.22008 article

L. Ángel and L. van den Dries: “Bounded pregeometries and pairs of fields,” South Am. J. Log. 2 : 2 (2016), pp. 459–475. For Francisco Miraglia, on his 70th birthday. MR 3671046 ArXiv 1707.03486 article

Francisco Miraglia	Related
Leonardo Ángel	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

M. Aschenbrenner, L. van den Dries, and J. van der Hoeven: “Dimension in the realm of transseries,” pp. 23–39 in Ordered algebraic structures and related topics (Luminy, France, 12–16 October 2015). Edited by F. Broglia, F. Delon, M. Dickmann, D. Gondard-Cozette, and V. A. Powers. Contemporary Mathematics 697. American Mathematical Society (Providence, RI), 2017. MR 3716064 Zbl 1388.12008 ArXiv 1607.07173 incollection

Françoise Delon	Related
Matthias Aschenbrenner	Related
Maximo Alejandro Dickmann	Related
Joris van der Hoeven	Related
Danielle Gondard-Cozette	Related
Victoria Ann Powers	Related
Fabrizio Broglia	Related

M. Aschenbrenner, L. van den Dries, and J. van der Hoeven: “Maximal immediate extensions of valued differential fields,” Proc. Lond. Math. Soc. (3) 117 : 2 (April 2018), pp. 376–406. MR 3851327 Zbl 06929623 article

Joris van der Hoeven	Related
Matthias Aschenbrenner	Related

L. van den Dries and N. Pynn-Coates: “On the uniqueness of maximal immediate extensions of valued differential fields,” J. Algebra 519 (2019), pp. 87–100. MR 3874517 Zbl 06988683 article

L. van den Dries and P. Ehrlich: “Homogeneous universal \( H \)-fields,” Proc. Amer. Math. Soc. 147 : 5 (2019), pp. 2231–2234. MR 3937696 Zbl 07046542 article

M. Aschenbrenner, L. van den Dries, and J. van der Hoeven: “The surreal numbers as a universal \( H \)-field,” J. Eur. Math. Soc. (JEMS) 21 : 4 (2019), pp. 1179–1199. MR 3941461 Zbl 1470.12004 article

L. van den Dries, J. van der Hoeven, and E. Kaplan: “Logarithmic hyperseries,” Trans. Amer. Math. Soc. 372 : 7 (2019), pp. 5199–5241. MR 4009458 Zbl 07110652 article

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