Abstract
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 \(m(m+1)\).
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Notes
- 1.
In case \(r=0\), the cone over the empty set is a point.
- 2.
For any commutative k-domain S, ideal \(L\subset S\), and \(\delta \in \mathrm{Der}_k(S)\), we have \(\delta (L^{n+1})\subset L^n\), \(n\ge 0\).
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Acknowledgements
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.
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Freudenburg, G. (2020). The Super-Rank of a Locally Nilpotent Derivation of a Polynomial Ring. In: Kuroda, S., Onoda, N., Freudenburg, G. (eds) Polynomial Rings and Affine Algebraic Geometry. PRAAG 2018. Springer Proceedings in Mathematics & Statistics, vol 319. Springer, Cham. https://doi.org/10.1007/978-3-030-42136-6_5
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DOI: https://doi.org/10.1007/978-3-030-42136-6_5
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