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
Abstract
People
BibTeX
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.
@article {key590820m,
AUTHOR = {Cherlin, Gregory and van den Dries,
Lou and Macintyre, Angus},
TITLE = {Decidability and undecidability theorems
for {PAC}-fields},
JOURNAL = {Bull. Am. Math. Soc. (N.S.)},
FJOURNAL = {American Mathematical Society. Bulletin.
New Series},
VOLUME = {4},
NUMBER = {1},
YEAR = {1981},
PAGES = {101--104},
DOI = {10.1090/S0273-0979-1981-14872-2},
NOTE = {MR:590820. Zbl:0466.12017.},
ISSN = {0273-0979},
}
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
People
BibTeX
@article {key663136m,
AUTHOR = {van den Dries, L. P. D. and Glass, A.
M. W. and Macintyre, Angus and Mekler,
Alan H. and Poland, John},
TITLE = {Elementary equivalence and the commutator
subgroup},
JOURNAL = {Glasgow Math. J.},
FJOURNAL = {Glasgow Mathematical Journal},
VOLUME = {23},
NUMBER = {2},
MONTH = {July},
YEAR = {1982},
PAGES = {115--117},
DOI = {10.1017/S0017089500004870},
NOTE = {MR:663136. Zbl:0504.03006.},
ISSN = {0017-0895},
}
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
People
BibTeX
@article {key689765m,
AUTHOR = {Macintyre, Angus and McKenna, Kenneth
and van den Dries, Lou},
TITLE = {Elimination of quantifiers in algebraic
structures},
JOURNAL = {Adv. Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {47},
NUMBER = {1},
MONTH = {January},
YEAR = {1983},
PAGES = {74--87},
DOI = {10.1016/0001-8708(83)90055-5},
NOTE = {MR:689765. Zbl:0531.03016.},
ISSN = {0001-8708},
}
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
Abstract
People
BibTeX
@article {key963007m,
AUTHOR = {B\'elair, L. and van den Dries, L. and
Macintyre, A.},
TITLE = {Elementary equivalence and codimension
in \$p\$-adic fields},
JOURNAL = {Manuscripta Math.},
FJOURNAL = {Manuscripta Mathematica},
VOLUME = {62},
NUMBER = {2},
MONTH = {June},
YEAR = {1988},
PAGES = {219--225},
DOI = {10.1007/BF01278980},
NOTE = {MR:963007. Zbl:0665.12027.},
ISSN = {0025-2611},
}
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
People
BibTeX
@article {key1048527m,
AUTHOR = {van den Dries, Lou and Macintyre, Angus},
TITLE = {The logic of {R}umely's local-global
principle},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal f\"ur die Reine und Angewandte
Mathematik. [Crelle's Journal]},
VOLUME = {1990},
NUMBER = {407},
YEAR = {1990},
PAGES = {33--56},
NOTE = {MR:1048527. Zbl:0703.13021.},
ISSN = {0075-4102},
}
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
People
BibTeX
@article {key1162433m,
AUTHOR = {Chatzidakis, Zo\'e and van den Dries,
Lou and Macintyre, Angus},
TITLE = {Definable sets over finite fields},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal f\"ur die Reine und Angewandte
Mathematik. [Crelle's Journal]},
VOLUME = {1992},
NUMBER = {427},
MONTH = {May},
YEAR = {1992},
PAGES = {107--135},
NOTE = {MR:1162433. Zbl:0759.11045.},
ISSN = {0075-4102},
}
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
Abstract
People
BibTeX
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].
@article {key1289495m,
AUTHOR = {van den Dries, Lou and Macintyre, Angus
and Marker, David},
TITLE = {The elementary theory of restricted
analytic fields with exponentiation},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {140},
NUMBER = {1},
MONTH = {July},
YEAR = {1994},
PAGES = {183--205},
DOI = {10.2307/2118545},
NOTE = {MR:1289495. Zbl:0837.12006.},
ISSN = {0003-486X},
}
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
Abstract
People
BibTeX
We use generalized power series to construct algebraically a nonstandard model of the theory of the real field with exponentiation. This model enables us to show the undefinability of the zeta function and certain non-elementary and improper integrals. We also use this model to answer a question of Hardy by showing that the compositional inverse to the function
\[ (\log x) (\log\log x) \]
is not asymptotic as \( x\to +\infty \) to a composition of semialgebraic functions, \( \log \) and \( \exp \) .
@article {key1610431m,
AUTHOR = {van den Dries, Lou and Macintyre, Angus
and Marker, David},
TITLE = {Logarithmic-exponential power series},
JOURNAL = {J. London Math. Soc. (2)},
FJOURNAL = {Journal of the London Mathematical Society.
Second Series},
VOLUME = {56},
NUMBER = {3},
YEAR = {1997},
PAGES = {417--434},
DOI = {10.1112/S0024610797005437},
NOTE = {MR:1610431. Zbl:0924.12007.},
ISSN = {0024-6107},
}
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
People
BibTeX
@incollection {key1930683m,
AUTHOR = {van den Dries, Lou},
TITLE = {An intermediate value property for first-order
differential polynomials},
BOOKTITLE = {Connections between model theory and
algebraic and analytic geometry},
EDITOR = {Macintyre, Angus},
SERIES = {Quaderni di Matematica},
NUMBER = {6},
PUBLISHER = {Aracne},
ADDRESS = {Rome},
YEAR = {2000},
PAGES = {95--105},
NOTE = {MR:1930683. Zbl:0994.26005.},
ISSN = {2420-8450},
ISBN = {9788879993128},
}
L. van den Dries, A. Macintyre, and D. Marker :
“Logarithmic-exponential series 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
Abstract
People
BibTeX
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.
@article {key1848569m,
AUTHOR = {van den Dries, Lou and Macintyre, Angus
and Marker, David},
TITLE = {Logarithmic-exponential series},
JOURNAL = {Ann. Pure Appl. Logic},
FJOURNAL = {Annals of Pure and Applied Logic},
VOLUME = {111},
NUMBER = {1--2},
MONTH = {July},
YEAR = {2001},
PAGES = {61--113},
DOI = {10.1016/S0168-0072(01)00035-5},
NOTE = {\textit{Proceedings of the international
conference ``{A}nalyse \& {L}ogique''}
(Mons, Belgium, 25--29 August 1997).
Issue edited by C. Finet and C. Michaux.
MR:1848569. Zbl:0998.12014.},
ISSN = {0168-0072},
}
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
People
BibTeX
@incollection {key2159715m,
AUTHOR = {van den Dries, L. and Speissegger, P.},
TITLE = {o-minimal preparation theorems},
BOOKTITLE = {Model theory and applications},
EDITOR = {B\'elair, L. and Chatzidakis, Z. and
D'Aquino, P. and Marker, D. and Otero,
M. and Point, F. and Wilkie, A.},
SERIES = {Quaderni di Matematica},
NUMBER = {11},
PUBLISHER = {Aracne},
ADDRESS = {Rome},
YEAR = {2002},
PAGES = {87--116},
NOTE = {(Ravello, Italy, 27 May--1 June 2002).
To Angus Macintyre, on his 60th birthday.
MR:2159715. Zbl:1081.03039.},
ISSN = {2420-8450},
ISBN = {9788879994118},
}
M. Aschenbrenner and L. van den Dries :
“Asymptotic differential algebra 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
Abstract
People
BibTeX
We believe there is room for a subject named as in the title of this paper. Motivating examples are Hardy fields and fields of transseries. Assuming no previous knowledge of these notions, we introduce both, state some of their basic properties, and explain connections to o-minimal structures. We describe a common algebraic framework for these examples: the category of \( H \) -fields . This unified setting leads to a better understanding of Hardy fields and transseries from an algebraic and model-theoretic perspective.
@incollection {key2130825m,
AUTHOR = {Aschenbrenner, Matthias and van den
Dries, Lou},
TITLE = {Asymptotic differential algebra},
BOOKTITLE = {Analyzable functions and applications},
EDITOR = {Costin, O. and Kruskal, M. D. and Macintyre,
A.},
SERIES = {Contemporary Mathematics},
NUMBER = {373},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2005},
PAGES = {49--85},
DOI = {10.1090/conm/373/06914},
NOTE = {(Edinburgh, UK, 17--21 June 2002). MR:2130825.
Zbl:1087.12002.},
ISSN = {0271-4132},
ISBN = {9780821834190},
}
