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

 

Super-additive sequences and algebras of polynomials


Author: Keith Johnson
Journal: Proc. Amer. Math. Soc. 139 (2011), 3431-3443
MSC (2010): Primary 13F20; Secondary 05A10, 11C08
Published electronically: March 4, 2011
MathSciNet review: 2813375
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Abstract: If $ K$ is a field with discrete valuation $ \nu$ and $ D= \{ a \in K: \nu (a) \geq 0\}$, then an algebra $ D[x]\subseteq A\subseteq K[x]$ has associated to it a sequence of fractional ideals $ \{{\mathcal I}_n:n=0,1,2,\dots\}$ with $ {\mathcal I}_n$ consisting of 0 and the leading coefficients of elements of $ A$ of degree no more than $ n$ and a sequence of integers $ \{a(n):n=0,1,2,\dots\}$ with $ a(n)=-\nu({\mathcal I}_n)$. Combinatorial properties of this integer sequence reflect algebraic properties of $ A$, and these are used to identify the degrees of generators of $ A$ and to characterize finitely generated algebras $ A$ by a periodicity property of this sequence.


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Additional Information

Keith Johnson
Affiliation: Department of Mathematics, Dalhousie University, Halifax, Nova Scotia, B3H 4R2, Canada
Email: johnson@mathstat.dal.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10785-8
PII: S 0002-9939(2011)10785-8
Received by editor(s): May 21, 2010
Received by editor(s) in revised form: August 30, 2010
Published electronically: March 4, 2011
Communicated by: Irena Peeva
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.