Weitzenb\"ock derivations of nilpotency 3

TL;DR

This paper studies the algebra of constants under a specific derivation with Jordan blocks of size at most 3, providing a minimal generating set and an algorithm for expressing invariants as polynomials in these generators.

Contribution

It extends previous work on derivations with Jordan blocks of size 2 to the case of size 3, offering a minimal generating set and an algorithm using combinatorial methods.

Findings

Provided a minimal generating set for the algebra of constants with Jordan blocks of size at most 3.

Developed an algorithm to express any invariant as a polynomial in the generators.

Demonstrated how classical polarization and restitution techniques can augment SAGBI basis methods.

Abstract

We consider a Weitzenb\"ock derivation $\Delta$ acting on a polynomial ring $R=K[\xi_1,\xi_2,...,\xi_m]$ over a field $K$ of characteristic 0. The $K$-algebra $R^\Delta = \{h \in R \mid \Delta(h) = 0\}$ is called the algebra of constants. Nowicki considered the case where the Jordan matrix for $\Delta$ acting on $R_1$, the degree 1 component of $R$, has only Jordan blocks of size 2. He conjectured (\cite{N}) that a certain set generates $R^{\Delta}$ in that case. Recently Koury (\cite{Kh}), Drensky and Makar-Limanov (\cite{DM}) and Kuroda (\cite{K}) have given proofs of Nowicki's conjecture. Here we consider the case where the Jordan matrix for $\Delta$ acting on $R_{1}$ 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 $R^{\Delta}$. Moreover, we show how our proof yields an algorithm to express any $h \in R^\Delta$ as a polynomial in the elements of $\mathcal G$. 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.