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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
DOI: https://doi.org/10.1090/S0002-9947-05-03820-1
Published electronically: January 21, 2005
MathSciNet review: 2171228
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Abstract | References | Similar Articles | Additional Information

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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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

DOI: https://doi.org/10.1090/S0002-9947-05-03820-1
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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