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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

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ISSN
1435-5337
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Volume 26, Issue 2

# Weitzenböck derivations of nilpotency 3

David L. Wehlau
Published Online: 2012-02-03 | DOI: https://doi.org/10.1515/forum-2011-0038

## Abstract

We consider a Weitzenböck derivation Δ acting on a polynomial ring $R=K\left[{\xi }_{1},{\xi }_{2},...,{\xi }_{m}\right]$ over a field K of characteristic 0. The K-algebra ${R}^{\Delta }=\left\{h\in R\mid \Delta \left(h\right)=0\right\}$ is called the algebra of constants. Nowicki considered the case where the Jordan matrix for Δ acting on R1, the degree 1 component of R, has only Jordan blocks of size 2. He conjectured that a certain set generates ${R}^{\Delta }$ in that case. Recently Khoury, Drensky and Makar-Limanov and Kuroda have given proofs of Nowicki's conjecture. Here we consider the case where the Jordan matrix for Δ acting on R1 has only Jordan blocks of size at most 3. We use combinatorial methods to give a minimal set of generators 𝒢 for the algebra of constants ${R}^{\Delta }$. Moreover, we show how our proof yields an algorithm to express any $h\in {R}^{\Delta }$ as a polynomial in the elements of 𝒢. In particular, our solution shows how the classical techniques of polarization and restitution may be used to augment the techniques of SAGBI bases to construct generating sets for subalgebras.

MSC: 13N15; 13A50; 13P10; 14E07

Revised: 2011-11-07

Published Online: 2012-02-03

Published in Print: 2014-03-01

Citation Information: Forum Mathematicum, Volume 26, Issue 2, Pages 577–591, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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