Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Well-Ordered Valuations on Rational Function Fields in Two Variables (1712.08325v2)

Published 22 Dec 2017 in math.AC

Abstract: Gr\"obner bases have been generalized by replacing monomial orders with constructions such as valuations and filtrations. We consider suitable valuations on a rational valuation field $K(x,y)$ and analyze their behavior when restricting to an underlying polynomial ring $K[x,y]$. In previous work, the corresponding value groups were subsets of ${\mathbb Q}$, and in this paper we consider the case when the value groups are isomorphic to ${\mathbb Z} \oplus {\mathbb Z}$. Bounds on how the image of $K[x,y]$ grows with respect to degree are given, and then a class a valuations that are suitable for use for generalized Gr\"obner bases are described. We construct an example in which the image of the underlying polynomial ring is non-negative, yet is not well-ordered.

Summary

We haven't generated a summary for this paper yet.