2010/11/01 by David L. Wehlau, Wehlau, David L.
Mathematics · #13A50 #13N15 #13P10 #14E07 #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA) #math.AC #math.RA #msc:13A50 #msc:13N15 #msc:13P10 #msc:14E07
paper · pdf · doi:10.48550/arxiv.1011.0454
added a short section (#9) which outlines a limitation on the technique used in the paper; simplified some of the exposition; corrected a number of typos. [15 pages]
arxiv created 2012/03/14 · arxiv updated 2012/03/16
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= \h ∈ R | Δ(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 (\citeN) that a certain set generates RΔ in that case. Recently Koury (\citeKh), Drensky and Makar-Limanov (\citeDM) and Kuroda (\citeK) 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. Here we use combinatorial methods to give a minimal set of generators \mathcal G for the algebra of constants RΔ. Moreover, we show how our proof yields an algorithm to express any h ∈ RΔ as a polynomial in the elements of \mathcal 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.