An algorithm to compute the kernel of a derivation up to a certain degree (Q5890234)

The well-known algorithm of \textit{A. van den Essen} [J. Symb. Comput. 16, 551-555 (1993; Zbl 0809.13002)] calculates generators for the kernel of a locally nilpotent \(k\)-derivation \(D\) of a commutative noetherian \(k\)-domain, where \(k\) is any field of characteristic zero. The algorithm terminates if, and only if, the kernel of \(D\) is finitely generated over \(k\). Its theoretical importance is seen, for example, in the proof that the kernel of a triangular derivation of a polynomial ring in four variables is always finitely generated [\textit{D. Daigle} and \textit{G. Freudenburg}, J. Algebra 241, No. 1, 328-339 (2001; Zbl 1018.13013)]. In practice, however, the algorithm is very time-consuming, due to its heavy dependence on Gröbner basis computations. NEWLINENEWLINENEWLINEIn the article under review, the author gives an alternative algorithm which is theoretically more narrow in scope, but in practice much faster. The reason for its enhanced speed is that it only needs to calculate kernels of vector space maps, rather than generators of ideals. Specifically, the author considers derivations \(D\) of polynomial rings \(k[x_1,\dots,x_n]\) (\(D\) need not be locally nilpotent); this is a very important case to understand. Maubach's algorithm calculates kernel generators up to degree \(n\) for any chosen \(n\in{\mathbb Z}^+\). Unlike van den Essen's algorithm, however, the new procedure does not determine whether the polynomials it produces form a complete set of generators for the kernel. But the author points out that this can be decided (very quickly) by the main step of van den Essen's algorithm. Moreover, for homogeneous derivations, it is shown that, if the algorithm computes a generating set for the kernel, then this set is minimal. NEWLINENEWLINENEWLINEOverall, this is a very nice paper, and provides a useful tool for researchers wanting to experiment with various derivations. The main idea is that when \(D\) is homogeneous of degree \(d\) with respect to one (or more) \({\mathbb N}\)-gradings of the polynomial ring \(A=\bigoplus A_i\), then \(D\) restricts to a vector space map \(A_i\rightarrow A_{i+d}\) for each \(i\), and \(\ker D\cap A_i = \ker(D|_{A_i})\).