The uniform companion for large differential fields of characteristic 0

Tressl, Marcus

doi:10.1090/S0002-9947-05-03981-4

The uniform companion for large differential fields of characteristic 0
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Trans. Amer. Math. Soc. 357 (2005), 3933-3951 Request permission

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

J. Böger, Modelltheorie von Körpern mit paarweise kommutierenden Derivationen. Diplomarbeit, Freiburg, September 1996.
Zoé Chatzidakis, Lou van den Dries, and Angus Macintyre, Definable sets over finite fields, J. Reine Angew. Math. 427 (1992), 107–135. MR 1162433
J. Denef and L. Lipshitz, Power series solutions of algebraic differential equations, Math. Ann. 267 (1984), no. 2, 213–238. MR 738249, DOI 10.1007/BF01579200
L. van den Dries and K. Schmidt, Bounds in the theory of polynomial rings over fields. A nonstandard approach, Invent. Math. 76 (1984), no. 1, 77–91. MR 739626, DOI 10.1007/BF01388493
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
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
Wilfrid Hodges, Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993. MR 1221741, DOI 10.1017/CBO9780511551574
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, DOI 10.1006/jsco.1999.0344
E. R. Kolchin, Differential algebra and algebraic groups, Pure and Applied Mathematics, Vol. 54, Academic Press, New York-London, 1973. MR 0568864
Tracey McGrail, The model theory of differential fields with finitely many commuting derivations, J. Symbolic Logic 65 (2000), no. 2, 885–913. MR 1771092, DOI 10.2307/2586576
David Pierce, Differential forms in the model theory of differential fields, J. Symbolic Logic 68 (2003), no. 3, 923–945. MR 2000487, DOI 10.2178/jsl/1058448448
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, DOI 10.1006/jabr.1997.7359
Florian Pop, Embedding problems over large fields, Ann. of Math. (2) 144 (1996), no. 1, 1–34. MR 1405941, DOI 10.2307/2118581
Alexander Prestel and Martin Ziegler, Model-theoretic methods in the theory of topological fields, J. Reine Angew. Math. 299(300) (1978), 318–341. MR 491852
Michael F. Singer, The model theory of ordered differential fields, J. Symbolic Logic 43 (1978), no. 1, 82–91. MR 495120, DOI 10.2307/2271951
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, DOI 10.4064/bc58-0-15
William 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 727801, DOI 10.2307/2273676
William H. Wheeler, Model complete theories of pseudo-algebraically closed fields, Ann. Math. Logic 17 (1979), no. 3, 205–226. MR 556892, DOI 10.1016/0003-4843(79)90008-1