Rings of nilpotent elements of monomial derivations on polynomial rings

Abstract

We study the rings $\text{Nil}D$ of nilpotent elements of R-derivations D on polynomial rings over a UFD R containing $\mathbb{Q}$. The main result of this paper determines $\text{Nil}D$ of the monomial derivations D on $R[x,y]$. Furthermore, as its application, we prove that, for each positive integer m, there exists a monomial R-derivation with a slice on $R[x,y,z]$ such that the minimal number of generators of its kernel equals m.

KEYWORDS:

• Derivations
• polynomial rings

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 13N15
• Secondary: 13B25

Acknowledgments

The authors would like to thank Professors Shigeru Kuroda and Takanori Nagamine for their helpful comments on this research.

Additional information

Funding

Research of the second author was partially supported by Grant-in-Aid for Scientific Research (C) (No. JP21K03200) from Japan Society for the Promotion of Science. This work was supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University.