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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

Online
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1435-5337
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Volume 26, Issue 2

Issues

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[ξ1,ξ2,...,ξm] over a field K of characteristic 0. The K-algebra RΔ={hRΔ(h)=0} 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 RΔ 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 RΔ. Moreover, we show how our proof yields an algorithm to express any hRΔ 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.

Keywords: Locally nilpotent derivations; algebra of constants; invariants of unitriangular transformations; Robert's isomorphism; polarization; restitution; SAGBI bases

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

About the article

Received: 2011-05-06

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, DOI: https://doi.org/10.1515/forum-2011-0038.

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