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

DOI:
https://doi.org/10.1090/S0002-9939-04-07456-8

Published electronically:
July 20, 2004

MathSciNet review:
2084067

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Abstract | References | Similar Articles | Additional Information

Abstract: Given a valuation on the function field , we examine the set of images of nonzero elements of the underlying polynomial ring under this valuation. For an arbitrary field , a Noetherian power series is a map that has Noetherian (i.e., reverse well-ordered) support. Each Noetherian power series induces a natural valuation on . 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 denote the images under the valuation of all nonzero polynomials of at most degree in the variable . We construct a bound for the growth of with respect to 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

Email:
emosteig@lmu.edu

**Moss Sweedler**

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

Email:
moss_sweedler@cornell.edu

DOI:
https://doi.org/10.1090/S0002-9939-04-07456-8

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