Probabilistic algorithms for computing resolvent representations of regular differential ideals

Abstract

In a previous article (Cluzeau and Hubert in Appl Algebra Eng Commun Comput 13(5):395–425, 2003), we proved the existence of resolvent representations for regular differential ideals. The present paper provides practical algorithms for computing such representations. We propose two different approaches. The first one uses differential characteristic decompositions whereas the second one proceeds by prolongation and algebraic elimination. Both constructions depend on the choice of a tuple over the differential base field and their success relies on the chosen tuple to be separating. The probabilistic aspect of the algorithms comes from this choice. To control it, we exhibit a family of tuples for which we can bound the probability that one of its element is separating.