It is well known that symbolic computation of derivations often leads to expressions of which some terms cancel in the end. The authors start with an example: a third-order differential operator the naïve computation of which requires 24 N 3 terms to be considered, but only 6 N 3 terms do not cancel, where N is the number of variables. The point of the paper is to reformulate the problem in the terminology of Lie and Hopf algebras by studying vector spaces of labeled trees. By applying a theorem of Milnor and Moore on the structure of Hopf algebras, the authors can compute a set of trees that does not contain the cancelled terms. The last section describes a parallel version of the algorithm.
The paper is an extended version of a previous conference contribution and does not contain new results. Even a misprint is not corrected: the ratio serial-to-parallel is still reversed. The authors make extensive use of theorems they have proven in former papers. People who work on simplifying expressions should be interested in these ideas, but readers must have a good knowledge of algebra.
