Well-Ordered Valuations on Rational Function Fields in Two Variables

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.