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Quantifier elimination for algebraic $D$-groups

Authors: Piotr Kowalski and Anand Pillay
Journal: Trans. Amer. Math. Soc. 358 (2006), 167-181
MSC (2000): Primary 32C38, 03C60
Published electronically: January 21, 2005
MathSciNet review: 2171228
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Abstract: We prove that if $G$ is an algebraic $D$-group (in the sense of Buium over a differentially closed field $(K,\partial)$ of characteristic $0$, then the first order structure consisting of $G$ together with the algebraic $D$-subvarieties of $G, G\times G,\dots$, has quantifier-elimination. In other words, the projection on $G^{n}$ of a $D$-constructible subset of $G^{n+1}$ is $D$-constructible. Among the consequences is that any finite-dimensional differential algebraic group is interpretable in an algebraically closed field.

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  • [B] A. Buium, Differential Algebraic Groups of Finite Dimension, Springer-Verlag (1992). MR 1176753 (93i:12010)
  • [Bor] A. Borel, Linear Algebraic Groups, Second edition, Springer-Verlag, 1991. MR 1102012 (92d:20001)
  • [Ch] Z. Chatzidakis, A note on canonical bases and one-based types in supersimple theories, preprint 2002.
  • [H1] E. Hrushovski, The Mordell-Lang conjecture for function fields, J. Amer. Math. Soc. 9 (1996), 667-690. MR 1333294 (97h:11154)
  • [H2] E. Hrushovski, Geometric Model Theory, Proceedings of ICM 1998, Vol. I, Documenta Math. 1998. MR 1648035 (2000b:03120)
  • [HP] E. Hrushovski and A. Pillay, Weakly normal groups, in Logic Colloquium '85, North-Holland, 1987. MR 0895647 (88e:03051)
  • [M] D. Marker, Model theory of differential fields. In M. Messmer, D. Marker and A. Pillay, editors, The Model Theory of Fields, vol. 5 of Lecture Notes in Logic. Springer-Verlag, 1996. MR 1477154 (98m:03075)
  • [PP] D. Pierce and A. Pillay, A Note on the Axioms for Differentially Closed Fields of Characteristic Zero, Journal of Algebra 204 (1998), 108-115. MR 1623945 (99g:12006)
  • [P1] A. Pillay, Some foundational questions concerning differential algebraic groups, Pacific Journal of Mathematics (1) 179 (1997), 179-200. MR 1452531 (98g:12008)
  • [P2] A. Pillay, Model-theoretic consequences of a theorem of Campana and Fujiki, Fundamenta Mathematicae, 174 (2002), 187-192. MR 1927236 (2003g:03057)
  • [P3] A. Pillay, Two remarks on differential fields, preprint, People/pillay/remarks.difffields.pdf
  • [P4] A. Pillay, Mordell-Lang for function fields in characteristic 0, revisited, Compositio Math. 140 (2004), 64-68. MR 2004123
  • [P5] A. Pillay, Geometric Stability Theory, Oxford University Press, 1996. MR 1429864 (98a:03049)
  • [P6] A. Pillay, Notes on analysability and canonical bases, preprint, People/pillay/remark.zoe.pdf
  • [PZ] A. Pillay, M. Ziegler, Jet spaces of varieties over differential and difference fields, Selecta Math., New Ser. 9 (2003), 579-599. MR 2031753 (2004m:12011)
  • [Sh] I. Shafarevich, Basic Algebraic Geometry 1, 2nd ed., Springer-Verlag, New York/Berlin 1994. MR 1328833 (95m:14001)
  • [Zi] B. Zilber, Model theory and algebraic geometry, Proceedings of 10th Easter Conference in Berlin, 1993, Humboldt Univ. of Berlin.

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Additional Information

Piotr Kowalski
Affiliation: Department of Mathematics, Wrocław University, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

Anand Pillay
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801

Received by editor(s): January 5, 2004
Published electronically: January 21, 2005
Additional Notes: The first author was supported by a postdoc under NSF Focused Research Grant DMS 01-00979 and the Polish KBN grant 2 P03A 018 24
The second author was partially supported by NSF grants DMS 00-70179 and DMS 01-00979
Article copyright: © Copyright 2005 American Mathematical Society

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