Tame and wild degree functions

Abstract

We give examples of degree functions $\deg\colon R \to M \cup \{-\infty\}$, where $R$ is $\mathbb{C}[X,Y]$ or $\mathbb{C}[X,Y,Z]$ and $M$ is $\mathbb{Z}$ or $\mathbb{N}$, whose behaviour with respect to $\mathbb{C}$-derivations $D\colon R \to R$ is pathological in the sense that $\{\deg(Dx) - \deg(x) \mid x \in R\setminus \{0\}\}$ is not bounded above. We also give several general results stating that such pathologies do not occur when the degree functions satisfy certain hypotheses.