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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Point count divisibility for algebraic sets over $ {\mathbb{Z}}/p^\ell{\mathbb{Z}}$ and other finite principal rings


Author: Daniel J. Katz
Journal: Proc. Amer. Math. Soc. 137 (2009), 4065-4075
MSC (2000): Primary 11T06; Secondary 13M10
Published electronically: July 28, 2009
MathSciNet review: 2538567
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Abstract | References | Similar Articles | Additional Information

Abstract: We determine the greatest common divisor of the cardinalities of the algebraic sets generated by collections of polynomials $ f_1,\ldots,f_t$ of specified degrees $ d_1,\ldots,d_t$ in $ n$ variables over a finite principal ring $ R$. This generalizes the theorems of Ax ($ t=1$, $ R$ a field), N. M. Katz ($ t$ arbitrary, $ R$ a field), and Marshall-Ramage ($ t=1$, $ R$ an arbitrary finite principal ring).


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Additional Information

Daniel J. Katz
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: katz.daniel.j@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10017-5
PII: S 0002-9939(09)10017-5
Received by editor(s): July 10, 2007
Received by editor(s) in revised form: April 26, 2009
Published electronically: July 28, 2009
Additional Notes: This work is in the public domain
Communicated by: Ted Chinburg