Tame and wild degree functions (Q411704)

In the article under review, the author gives examples of integer valued degree functions \(\deg\) on polynomial rings in two and three variables which are badly behaved with respect to derivations \(D\) of these rings, in the sense that the difference \(\deg(D(x))-\deg(x)\) is not bounded above. This prevents in particular such derivations to induce homogeneous derivations of the graded ring associated with the degree function.NEWLINENEWLINEThese examples illustrate the fact that the existence of associated homogeneous derivations, in particular in the locally nilpotent case, has been sometimes slightly overlooked in various preprints and published articles. But the author also lists several characterizations of when such pathologies do not occur. These confirm in particular that the classical and frequently used arguments via reduction to homogeneous locally nilpotent derivations are legitimate, even though they were sometimes justified either inaccurately or too quickly.