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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Factorial and Noetherian subrings of power series rings
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by Damek Davis and Daqing Wan PDF
Proc. Amer. Math. Soc. 139 (2011), 823-834 Request permission

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
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2745635