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Proceedings of the American Mathematical Society

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Factorial and Noetherian subrings of power series rings


Authors: Damek Davis and Daqing Wan
Journal: Proc. Amer. Math. Soc. 139 (2011), 823-834
MSC (2010): Primary 13F25, 14A05; Secondary 14F30
DOI: https://doi.org/10.1090/S0002-9939-2010-10620-2
Published electronically: August 6, 2010
MathSciNet review: 2745635
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Abstract: Let $F$ be a field. We show that certain subrings contained between the polynomial ring $F[X] = F[X_1, \cdots , X_n]$ and the power series ring $F[X][[Y]] = F[X_1, \cdots , X_n][[ Y]]$ have Weierstrass Factorization, which allows us to deduce both unique factorization and the Noetherian property. These intermediate subrings are obtained from elements of $F[X][[ Y]]$ by bounding their total $X$-degree above by a positive real-valued monotonic up function $\lambda$ on their $Y$-degree. These rings arise naturally in studying the $p$-adic analytic variation of zeta functions over finite fields. Future research into this area may study more complicated subrings in which $Y = (Y_1, \cdots , Y_m)$ has more than one variable, and for which there are multiple degree functions, $\lambda _1, \cdots , \lambda _m$. Another direction of study would be to generalize these results to $k$-affinoid algebras.


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Additional Information

Damek Davis
Affiliation: Department of Mathematics, University of California, Irvine, California 92697-3875
Address at time of publication: Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095-1555
Email: davisds@uci.edu, damek@math.ucla.edu

Daqing Wan
Affiliation: Department of Mathematics, University of California, Irvine, California 92697-3875
MR Author ID: 195077
Email: dwan@math.uci.edu

Received by editor(s): October 22, 2009
Received by editor(s) in revised form: April 8, 2010
Published electronically: August 6, 2010
Additional Notes: The second author is partially supported by the NSF
Communicated by: Ted Chinburg
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.