The uniform companion for large differential fields of characteristic 0

Author:
Marcus Tressl

Journal:
Trans. Amer. Math. Soc. **357** (2005), 3933-3951

MSC (2000):
Primary 03C65, 12H05; Secondary 03C10, 13N99

DOI:
https://doi.org/10.1090/S0002-9947-05-03981-4

Published electronically:
May 10, 2005

MathSciNet review:
2159694

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Abstract: We show that there is a theory UC of differential fields (in several commuting derivatives) of characteristic , which serves as a model companion for every theory of large and differential fields extending a model complete theory of pure fields. As an application, we introduce differentially closed ordered fields, differentially closed p-adic fields and differentially closed pseudo-finite fields.

**[Bö]**J. Böger,*Modelltheorie von Körpern mit paarweise kommutierenden Derivationen*. Diplomarbeit, Freiburg, September 1996.**[CDM]**Zoé Chatzidakis, Lou van den Dries, and Angus Macintyre,*Definable sets over finite fields*, J. Reine Angew. Math.**427**(1992), 107–135. MR**1162433****[DL]**J. Denef, L. Lipshitz,*Power Series Solutions of Algebraic Differential Equations*. Math. Ann. 267 (1984), 213-238. MR**0738249 (85j:12010)****[vdD-Sch]**L. van den Dries, K. Schmidt,*Bounds in the theory of polynomial rings over fields. A nonstandard approach*. Invent. Math. 76 (1984), no. 1, 77-91. MR**0739626 (85i:12016)****[E]**David Eisenbud,*Commutative algebra*, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR**1322960****[GaMi]**Giovanni Gallo and Bhubaneswar Mishra,*Efficient algorithms and bounds for Wu-Ritt characteristic sets*, Effective methods in algebraic geometry (Castiglioncello, 1990) Progr. Math., vol. 94, Birkhäuser Boston, Boston, MA, 1991, pp. 119–142. MR**1106418****[Ho]**Wilfrid Hodges,*Model theory*, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993. MR**1221741****[Hu]**Evelyne Hubert,*Factorization-free decomposition algorithms in differential algebra*, J. Symbolic Comput.**29**(2000), no. 4-5, 641–662. Symbolic computation in algebra, analysis, and geometry (Berkeley, CA, 1998). MR**1769659**, https://doi.org/10.1006/jsco.1999.0344**[Ko]**E. R. Kolchin,*Differential algebra and algebraic groups*, Academic Press, New York-London, 1973. Pure and Applied Mathematics, Vol. 54. MR**0568864****[McG]**Tracey McGrail,*The model theory of differential fields with finitely many commuting derivations*, J. Symbolic Logic**65**(2000), no. 2, 885–913. MR**1771092**, https://doi.org/10.2307/2586576**[Pi]**David Pierce,*Differential forms in the model theory of differential fields*, J. Symbolic Logic**68**(2003), no. 3, 923–945. MR**2000487**, https://doi.org/10.2178/jsl/1058448448**[PP]**David Pierce and Anand Pillay,*A note on the axioms for differentially closed fields of characteristic zero*, J. Algebra**204**(1998), no. 1, 108–115. MR**1623945**, https://doi.org/10.1006/jabr.1997.7359**[Po]**Florian Pop,*Embedding problems over large fields*, Ann. of Math. (2)**144**(1996), no. 1, 1–34. MR**1405941**, https://doi.org/10.2307/2118581**[PZ]**A. Prestel, M. Ziegler,*Model-theoretic methods in the theory of topological fields*. J. Reine Angew. Math. 299(300) (1978), 318-341. MR**0491852 (80f:54034)****[Si]**M. Singer,*The Model Theory of Ordered Differential Fields*. The Journal of Symbolic Logic, Vol. 43, Number 1, March 1978, pp. 82-91. MR**0495120 (80a:03044)****[Tr]**Marcus Tressl,*A structure theorem for differential algebras*, Differential Galois theory (Będlewo, 2001) Banach Center Publ., vol. 58, Polish Acad. Sci. Inst. Math., Warsaw, 2002, pp. 201–206. MR**1972455**, https://doi.org/10.4064/bc58-0-15**[Wh1]**W. H. Wheeler,*Model complete theories of formally real fields and formally -adic fields*. J. Symbolic Logic 48 (1983), no. 4, 1130-1139 (1984). MR**0727801 (85b:03055)****[Wh2]**W. H. Wheeler,*Model complete theories of pseudo-algebraically closed fields*. Ann. Math. Logic 17 (1979), no. 3, 205-226. MR**0556892 (81c:03024)**

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Additional Information

**Marcus Tressl**

Affiliation:
NWF-I Mathematik, 93040 Universität Regensburg, Germany

Email:
marcus.tressl@mathematik.uni-regensburg.de

DOI:
https://doi.org/10.1090/S0002-9947-05-03981-4

Keywords:
Differential algebra,
differentially closed,
large field,
model theory,
model complete

Received by editor(s):
April 22, 2003

Published electronically:
May 10, 2005

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.