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The growth of valuations on rational function fields in two variables

Authors: Edward Mosteig and Moss Sweedler
Journal: Proc. Amer. Math. Soc. 132 (2004), 3473-3483
MSC (2000): Primary 13F30, 13F25; Secondary 13P10
Published electronically: July 20, 2004
MathSciNet review: 2084067
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Abstract: Given a valuation on the function field $k(x,y)$, we examine the set of images of nonzero elements of the underlying polynomial ring $k[x,y]$ under this valuation. For an arbitrary field $k$, a Noetherian power series is a map $z:\mathbb{Q}\to k$ that has Noetherian (i.e., reverse well-ordered) support. Each Noetherian power series induces a natural valuation on $k(x,y)$. Although the value groups corresponding to such valuations are well-understood, the restrictions of the valuations to underlying polynomial rings have yet to be characterized. Let $\Lambda_n$ denote the images under the valuation $v$ of all nonzero polynomials $f \in k[x,y]$of at most degree $n$ in the variable $y$. We construct a bound for the growth of $\Lambda_n$ with respect to $n$ for arbitrary valuations, and then specialize to valuations that arise from Noetherian power series. We provide a sufficient condition for this bound to be tight.

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Additional Information

Edward Mosteig
Affiliation: Department of Mathematics, Loyola Marymount University, Los Angeles, California 90045

Moss Sweedler
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853

Keywords: Valuations, generalized power series, Gr\"obner bases
Received by editor(s): January 10, 2002
Received by editor(s) in revised form: July 14, 2003
Published electronically: July 20, 2004
Communicated by: Michael Stillman
Article copyright: © Copyright 2004 American Mathematical Society

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