2017/12/22 by Mosteig, Edward
#13A18 #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1712.08325
Gröbner 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 ⊕ \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 Gröbner bases are described. We construct an example in which the image of the underlying polynomial ring is non-negative, yet is not well-ordered.