Reversely well-ordered valuations on rational function fields in two variables

Abstract

We examine valuations on a rational function field K(x, y) and analyze their behavior when restricting to an underlying polynomial ring $K[x,y]$. Motivated to solve the ideal membership problem in polynomial rings using Moss Sweedler’s framework of generalized Gröbner bases, we produce an infinite collection of valuations $v:K(x,y)\to \mathbb{Z}\oplus \mathbb{Z}$ such that $v(K{[x,y]}^{*})$ is reversely well ordered. In addition, we construct a surprising example where $v(K{[x,y]}^{*})$ is nonpositive, yet not reversely well ordered.

Keywords:

• Valuations
• Gröbner bases

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• MSC2010: 13P10
• 13A18
• 13F30
• 68W30

Acknowledgements

We would like to express our gratitude to the participants of the Workshop on “Valuations on Rational Function Fields” held in the Department of Mathematics and Physics at the University of Szczecin in May of 2018 for the many helpful corrections and suggestions for improvement.