Super-additive sequences and algebras of polynomials
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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.References
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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
- 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
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 3431-3443
- MSC (2010): Primary 13F20; Secondary 05A10, 11C08
- DOI: https://doi.org/10.1090/S0002-9939-2011-10785-8
- MathSciNet review: 2813375