The Super-Rank of a Locally Nilpotent Derivation of a Polynomial Ring

The super-rank of a k-derivation of a polynomial ring \(k^{[n]}\) over a field k of characteristic zero is introduced. Like rank, super-rank is invariant under conjugation, and thus gives a way to classify derivations of maximal rank n. For each \(m\ge 2\), we construct a locally nilpotent derivation of \(k^{[m(m+1)]}\) with maximal super-rank \(m(m+1)\).

1. 1.

   In case \(r=0\), the cone over the empty set is a point.

2. 2.

   For any commutative k-domain S, ideal \(L\subset S\), and \(\delta \in \mathrm{Der}_k(S)\), we have \(\delta (L^{n+1})\subset L^n\), \(n\ge 0\).

The author wishes to thank Steve Mackey of Western Michigan University for his comments about an earlier version of this paper which led to a number of improvements in the final version.

Authors and Affiliations

1. Department of Mathematics, Western Michigan University, Kalamazoo, MI, 49008, USA

   Gene Freudenburg

Corresponding author

Correspondence to Gene Freudenburg .

Editors and Affiliations

1. Department of Mathematical Sciences, Tokyo Metropolitan University, Hachioji, Tokyo, Japan

   Shigeru Kuroda

2. Faculty of Applied Physics, University of Fukui, Fukui, Japan

   Nobuharu Onoda

3. Department of Mathematics, Western Michigan University, Kalamazoo, MI, USA

   Gene Freudenburg

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