Tame and wild degree functions

TL;DR

This paper explores unusual degree functions on polynomial rings that exhibit unbounded behavior under derivations, and establishes conditions under which such pathologies are avoided.

Contribution

It introduces examples of degree functions with pathological derivation behavior and provides criteria preventing these anomalies.

Findings

Examples of degree functions with unbounded derivation behavior.

Conditions under which degree functions behave normally.

Insights into the structure of polynomial rings and derivations.

Abstract

We give examples of degree functions deg : R --> M, where R is a polynomial ring in 2 or 3 variables and M is either the integers or the natural numbers, whose behaviour with respect to derivations D : R --> R is pathological in the sense that deg(Dx) - deg(x) (where x in R \ {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.