The growth of valuations on rational function fields in two variables
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- by Edward Mosteig and Moss Sweedler PDF
- Proc. Amer. Math. Soc. 132 (2004), 3473-3483 Request permission
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.References
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Additional Information
- Edward Mosteig
- Affiliation: Department of Mathematics, Loyola Marymount University, Los Angeles, California 90045
- Email: emosteig@lmu.edu
- Moss Sweedler
- Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
- Email: moss_sweedler@cornell.edu
- 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
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 3473-3483
- MSC (2000): Primary 13F30, 13F25; Secondary 13P10
- DOI: https://doi.org/10.1090/S0002-9939-04-07456-8
- MathSciNet review: 2084067