Weitzenböck derivations of nilpotency 3

Abstract

Abstract We consider a Weitzenböck derivation Δ acting on a polynomial ring R = K [ ξ 1 , ξ 2 , ... , ξ m ] $ R=K[\xi _1,\xi _2,\ldots ,\xi _m] $ over a field K of characteristic 0. The K-algebra R Δ = { h ∈ R ∣ Δ ( h ) = 0 } ${R^\Delta = \lbrace h \in R \mid \Delta (h) = 0\rbrace }$ is called the algebra of constants. Nowicki considered the case where the Jordan matrix for Δ acting on R 1, the degree 1 component of R, has only Jordan blocks of size 2. He conjectured that a certain set generates R Δ $R^{\Delta }$ 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 R 1 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 Δ $R^{\Delta }$ . Moreover, we show how our proof yields an algorithm to express any h ∈ R Δ $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.