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
Abstract
People
BibTeX
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.
@article {key1743664m,
AUTHOR = {Aschenbrenner, Matthias and van den
Dries, Lou},
TITLE = {Closed asymptotic couples},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {225},
NUMBER = {1},
MONTH = {March},
YEAR = {2000},
PAGES = {309--358},
DOI = {10.1006/jabr.1999.8128},
NOTE = {MR:1743664. Zbl:0974.12015.},
ISSN = {0021-8693},
}
M. Aschenbrenner and L. van den Dries :
“\( H \) -fields and their Liouville extensionsMath. Z.
242 : 3
(2002 ),
pp. 543–588 .
MR
1985465 Zbl
1066.12002 article
Abstract
People
BibTeX
We introduce \( H \) -fields as ordered differential fields of a certain kind. Hardy fields extending \( \mathbb{R} \) , as well as the field of logarithmic-exponential series over \( \mathbb{R} \) are \( H \) -fields. We study Liouville extensions in the category of \( H \) -fields, as a step towards a model theory of \( H \) -fields. The main result is that an \( H \) -field has at most two Liouville closures.
@article {key1985465m,
AUTHOR = {Aschenbrenner, Matthias and van den
Dries, Lou},
TITLE = {\$H\$-fields and their {L}iouville extensions},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {242},
NUMBER = {3},
YEAR = {2002},
PAGES = {543--588},
DOI = {10.1007/s002090000358},
NOTE = {MR:1985465. Zbl:1066.12002.},
ISSN = {0025-5874},
}
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
Abstract
People
BibTeX
\( H \) -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.
@article {key2123981m,
AUTHOR = {Aschenbrenner, Matthias and van den
Dries, Lou},
TITLE = {Liouville closed \$H\$-fields},
JOURNAL = {J. Pure Appl. Algebra},
FJOURNAL = {Journal of Pure and Applied Algebra},
VOLUME = {197},
NUMBER = {1--3},
MONTH = {May},
YEAR = {2005},
PAGES = {83--139},
DOI = {10.1016/j.jpaa.2004.08.009},
NOTE = {MR:2123981. Zbl:1134.12004.},
ISSN = {0022-4049},
}
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},
}
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
Abstract
People
BibTeX
\( H \) -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.
@article {key2183848m,
AUTHOR = {Aschenbrenner, Matthias and van den
Dries, Lou and van der Hoeven, Joris},
TITLE = {Differentially algebraic gaps},
JOURNAL = {Selecta Math. (N.S.)},
FJOURNAL = {Selecta Mathematica. New Series},
VOLUME = {11},
NUMBER = {2},
MONTH = {December},
YEAR = {2005},
PAGES = {247--280},
DOI = {10.1007/s00029-005-0010-0},
NOTE = {MR:2183848. Zbl:1151.12002.},
ISSN = {1022-1824},
}
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
Abstract
People
BibTeX
The differential field of transseries extends the field of real Laurent series and occurs in various contexts: asymptotic expansions, analytic vector fields, and o-minimal structures, to name a few. We give an overview of the algebraic and model-theoretic aspects of this differential field and report on our efforts to understand its elementary theory.
@article {key3091660m,
AUTHOR = {Aschenbrenner, Matthias and van den
Dries, Lou and van der Hoeven, Joris},
TITLE = {Toward a model theory for transseries},
JOURNAL = {Notre Dame J. Form. Log.},
FJOURNAL = {Notre Dame Journal of Formal Logic},
VOLUME = {54},
NUMBER = {3--4},
YEAR = {2013},
PAGES = {279--310},
DOI = {10.1215/00294527-2143898},
NOTE = {For Anand Pillay, on his 60th birthday.
MR:3091660. Zbl:1314.03037.},
ISSN = {0029-4527},
}
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
People
BibTeX
@book {key3585498m,
AUTHOR = {Aschenbrenner, Matthias and van den
Dries, Lou and van der Hoeven, Joris},
TITLE = {Asymptotic differential algebra and
model theory of transseries},
SERIES = {Annals of Mathematics Studies},
NUMBER = {195},
PUBLISHER = {Princeton University Press},
YEAR = {2017},
PAGES = {xxi+849},
DOI = {10.1515/9781400885411},
NOTE = {MR:3585498. Zbl:06684722.},
ISSN = {0066-2313},
ISBN = {9780691175430},
}
M. Aschenbrenner, L. van den Dries, and J. van der Hoeven :
“Dimension in the realm of transseries 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
Abstract
People
BibTeX
Let \( \mathbb{T} \) be the differential field of transseries. We establish some
basic properties of the dimension of a definable subset of \( \mathbb{T}^n \) , also in relation to its codimension in the ambient space \( \mathbb{T}^n \) . The case of dimension 0 is of special interest, and can be characterized both in topological terms (discreteness) and in terms of the Herwig–Hrushovski–Macpherson notion of co-analyzability.
@incollection {key3716064m,
AUTHOR = {Aschenbrenner, Matthias and van den
Dries, Lou and van der Hoeven, Joris},
TITLE = {Dimension in the realm of transseries},
BOOKTITLE = {Ordered algebraic structures and related
topics},
EDITOR = {Broglia, Fabrizio and Delon, Fran\c{c}oise
and Dickmann, Max and Gondard-Cozette,
Danielle and Powers, Victoria Ann},
SERIES = {Contemporary Mathematics},
NUMBER = {697},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2017},
PAGES = {23--39},
DOI = {10.1090/conm/697/14044},
NOTE = {(Luminy, France, 12--16 October 2015).
ArXiv:1607.07173. MR:3716064. Zbl:1388.12008.},
ISSN = {0271-4132},
ISBN = {9781470429669},
}
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
Abstract
People
BibTeX
@article {key3851327m,
AUTHOR = {Aschenbrenner, Matthias and van den
Dries, Lou and van der Hoeven, Joris},
TITLE = {Maximal immediate extensions of valued
differential fields},
JOURNAL = {Proc. Lond. Math. Soc. (3)},
FJOURNAL = {Proceedings of the London Mathematical
Society. Third Series},
VOLUME = {117},
NUMBER = {2},
MONTH = {April},
YEAR = {2018},
PAGES = {376--406},
DOI = {10.1112/plms.12128},
NOTE = {MR:3851327. Zbl:06929623.},
ISSN = {0024-6115},
}
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
Abstract
People
BibTeX
@article {key3941461m,
AUTHOR = {Aschenbrenner, Matthias and van den
Dries, Lou and van der Hoeven, Joris},
TITLE = {The surreal numbers as a universal \$H\$-field},
JOURNAL = {J. Eur. Math. Soc. (JEMS)},
FJOURNAL = {Journal of the European Mathematical
Society (JEMS)},
VOLUME = {21},
NUMBER = {4},
YEAR = {2019},
PAGES = {1179--1199},
DOI = {10.4171/JEMS/858},
NOTE = {MR:3941461. Zbl:1470.12004.},
ISSN = {1435-9855},
}
