Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that there is a theory UC of differential fields (in several commuting derivatives) of characteristic $0$, 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.


References [Enhancements On Off] (What's this?)

  • [Bö] J. Böger, Modelltheorie von Körpern mit paarweise kommutierenden Derivationen. Diplomarbeit, Freiburg, September 1996.
  • [CDM] Z. Chatzidakis, L. van den Dries, A. Macintyre, Definable sets over finite fields. J. reine angew. Math. 427 (1992), 107-135. MR 1162433 (94c:03049)
  • [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] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry. Springer Graduate Texts in Mathematics 150, 1995. MR 1322960 (97a:13001)
  • [GaMi] G. Gallo, B. Mishra, Efficient Algorithms and Bounds for the Wu-Ritt Characteristic Sets. Effective methods in algebraic geometry (Castiglioncello, 1990), 119-142, Progr. Math., 94, Birkhäuser-Boston, Boston, MA, 1991. MR 1106418 (92e:14058)
  • [Ho] W. Hodges, Model Theory. Encyclopedia of Mathematics and its Applications, vol. 42, 1993. MR 1221741 (94e:03002)
  • [Hu] E. Hubert, Factorization-free Decomposition Algorithms in Differential Algebra. J. Symbolic Computation 29 (2000), 641-662. MR 1769659 (2001k:12013)
  • [Ko] E. R. Kolchin, Differential Algebra and Algebraic Groups. Pure and Applied Mathematics, vol. 54, Academic Press, 1973. MR 0568864 (58:27929)
  • [McG] T. McGrail, The model theory of differential fields with finitely many commuting derivations. J. Symbolic Logic 65 (2000), no. 2, 885-913. MR 1771092 (2001h:03066)
  • [Pi] D. Pierce, Differential Forms in the Model Theory of Differential Fields. J. Symbolic Logic 68 (2003), no. 3, 923-945. MR 2000487 (2004h:03080)
  • [PP] D. Pierce, A. Pillay, A note on the axioms for differentially closed fields of characteristic zero. J. Algebra 204 (1998), no. 1, 108-115. MR 1623945 (99g:12006)
  • [Po] F. Pop, Embedding problems over large fields. Ann. of Math. (2) 144 (1996), no. 1, 1-34. MR 1405941 (97h:12013)
  • [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] M. Tressl, A Structure Theorem for Differential Algebras. Banach Center Publications, vol. 58, Warszawa 2002, pp. 201-206. MR 1972455 (2004b:12011)
  • [Wh1] W. H. Wheeler, Model complete theories of formally real fields and formally $p$-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)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 03C65, 12H05, 03C10, 13N99

Retrieve articles in all journals with MSC (2000): 03C65, 12H05, 03C10, 13N99


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.

American Mathematical Society