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FIELDS WITH SEVERAL COMMUTING DERIVATIONS

Published online by Cambridge University Press:  17 April 2014

DAVID PIERCE
DAVID PIERCE*
Affiliation:
MATHEMATICS DEPARTMENT, MIMAR SINAN FINE ARTS UNIVERSITY, ISTANBUL, TURKEY E-mail: dpierce@msgsu.edu.tr, URL: http://mat.msgsu.edu.tr/∼dpierce
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Abstract

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For every natural number m, the existentially closed models ofthe theory of fields with m commuting derivations can be givena first-order geometric characterization in several ways. In particular, thetheory of these differential fields has a model-companion. The axioms are thatcertain differential varieties determined by certain ordinary varieties arenonempty. There is no restriction on the characteristic of the underlyingfield.

Keywords

3C10 (03C60, 12H05)Model-companionmodel-completionexistentially closed modelfield with several commuting derivationsderivative

Information

Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 1 , March 2014, pp. 1 - 19
Copyright
Copyright © Association for Symbolic Logic 2014 

References

REFERENCES

Barwise, Jon and Robinson, Abraham, Completing theories by forcing. Annals of Mathematical Logic, vol. 2 (1970), no. 2, pp. 119–142. MR MR0272613 (42 #7494) CrossRefGoogle Scholar
Blum, Lenore, Differentially closed fields: A model-theoretic tour, Contributions to algebra (collection of papers dedicated to Ellis Kolchin), Academic Press, New York, 1977, pp. 37–61. MR MR0491149 (58 #10415) Google Scholar
Chang, Chen Chung, On unions of chains of models. Proceedings of the American Mathematical Society, vol. 10 (1959), pp. 120–127. MR MR0103812 (21 #2576) Google Scholar
Chatzidakis, Zoé and Hrushovski, Ehud, Model theory of difference fields. Transactions of the American Mathematical Society, vol. 351 (1999), no. 8, pp. 2997–3071. MR2000f:03109 Google Scholar
Eklof, Paul and Sabbagh, Gabriel, Model-completions and modules. Annals of Mathematical Logic, vol. 2 (1970/1971), no. 3, pp. 251–295. MR MR0277372 (43 #3105) Google Scholar
Hodges, Wilfrid, Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993. MR94e:03002 Google Scholar
Kolchin, E. R., Differential algebra and algebraic groups, Academic Press, New York, 1973, Pure and Applied Mathematics, vol. 54. MR58 #27929 Google Scholar
Kowalski, Piotr, Derivations of the Frobenius map, this Journal, vol. 70 (2005), no. 1, pp. 99–110. MRMR2119125 (2005m:03069) Google Scholar
Łoś, Jerzy and Suszko, Roman, On the extending of models (IV): Infinite sums of models. Fundamenta Mathematicae, vol. 44 (1957), pp. 52–60. MR MR0089813 (19,724c) Google Scholar
Macintyre, Angus, Generic automorphisms of fields. Annals of Pure and Applied Logic, vol. 88 (1997), no. 2–3, pp. 165–180, Joint AILA-KGS Model Theory Meeting (Florence, 1995). MR99c:03046 CrossRefGoogle Scholar
Marker, David, Model theory: An introduction, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002. MR1 924 282 Google Scholar
McGrail, Tracey, The model theory of differential fields with finitely many commuting derivations, this Journal, vol. 65 (2000), no. 2, pp. 885–913. MR2001h:03066 Google Scholar
Pierce, David, Differential forms in the model theory of differential fields, this Journal, vol. 68 (2003), no. 3, pp. 923–945. MR2 000 487 Google Scholar
Pierce, David, Geometric characterizations of existentially closed fields with operators. Illinois Journal of Mathematics, vol. 48 (2004), no. 4, pp. 1321–1343. MRMR2114160 Google Scholar
Pierce, David and Pillay, Anand, A note on the axioms for differentially closed fields of characteristic zero. Journal of Algebra, vol. 204 (1998), no. 1, pp. 108–115. MR99g:12006 Google Scholar
Fels Ritt, Joseph, Differential algebra, Dover Publications Inc., New York, 1966, originally published in 1950. MRMR0201431 (34 #1315) Google Scholar
Robinson, A., Some problems of definability in the lower predicate calculus. Fundamenta Mathematicae, vol. 44 (1957), pp. 309–329. MRMR0091922 (19, 1032f) Google Scholar
Robinson, Abraham, Introduction to model theory and to the metamathematics of algebra, North-Holland Publishing Co., Amsterdam, 1963. MRMR0153570 (27 #3533) Google Scholar
Robinson, Abraham, Complete theories, second ed., North-Holland Publishing Co., Amsterdam, 1977, With a preface by H. J. Keisler, Studies in Logic and the Foundations of Mathematics, first published 1956. MRMR0472504 (57 #12202) Google Scholar
Rosenfeld, Azriel, Specializations in differential algebra. Transactions of the American Mathematical Society, vol. 90 (1959), pp. 394–407. MRMR0107642 (21 #6367) Google Scholar
Sacks, Gerald E., Saturated model theory, W. A. Benjamin, Inc., Reading, Massachusetts, 1972, Mathematics Lecture Note Series. MRMR0398817 (53 #2668) Google Scholar
Seidenberg, A., Some basic theorems in differential algebra (characteristic p, arbitrary). Transactions of the American Mathematical Society, vol. 73 (1952), pp. 174–190. MRMR0049174 (14,130c) Google Scholar
Seidenberg, A., An elimination theory for differential algebra, Berkeley, Los Angeles, University of California Publications in Mathematics (new series), vol. 3, 1956, pp. 31–65. MRMR0082487 (18,558a) Google Scholar
Shelah, Saharon, Differentially closed fields. Israel Journal of Mathematics, vol. 16 (1973),pp. 314–328. MRMR0344116 (49 #8856) Google Scholar
Simmons, H., Existentially closed structures, this Journal, vol. 37 (1972), pp. 293–310. MR51 #12518 Google Scholar
Singer, Michael F., The model theory of ordered differential fields, this Journal, vol. 43 (1978), no. 1, pp. 82–91. MRMR495120 (80a:03044) Google Scholar
Singer, Michael F., Model theory of partial differential fields: From commuting to noncommuting derivations. Proceedings of the American Mathematical Society, vol. 135 (2007), no. 6, pp. 1929–1934 (electronic). MRMR2286106 (2007m:03077) Google Scholar
Tressl, Marcus, The uniform companion for large differential fields of characteristic 0. Transactions of the American Mathematical Society, vol. 357 (2005), no. 10, pp. 3933–3951 (electronic).MRMR2159694 Google Scholar
Den Dries, L. Van and Schmidt, K., Bounds in the theory of polynomial rings over fields. A nonstandard approach. Inventiones Mathematicae, vol. 76 (1984), no. 1, pp. 77–91. MRMR739626 (85i:12016) Google Scholar
Wood, Carol, The model theory of differential fields of characteristic P≠0. Proceedings of the American Mathematical Society , vol. 40 (1973), pp. 577–584. MR48 #8227 Google Scholar
Wood, Carol, Prime model extensions for differential fields of characteristic P≠0, this Journal, vol. 39 (1974), pp. 469–477. MR50 #9577 Google Scholar
Yaffe, Yoav, Model completion of Lie differential fields. Annals of Pure and Applied Logic, vol. 107 (2001), no. 1–3, pp. 49–86. MR1 807 840 CrossRefGoogle Scholar