Computing canonical representatives of regular differential ideals
François Boulier, François Lemaire · 2000
In this paper, we give three theoretical and practical contributions for solving polynomial ODE or PDE systems. The first one is practical: an algorithm which improves the purely algebraic part of Rosenfeld—Gröbner (the polynomial ODE or PDE systems simplifier which is the core of the Maple 5.5 diffalg package). It is a variant of lextriangular but does not need any Gröbner basis computation. The second one is theoretical: a characterization of the output of Rosenfeld—Gröbner and a clarification of the existing relationship between algebraic and differential characteristic sets. The third one is theoretical as well as practical: an algorithm to compute canonical representatives of differential polynomials modulo regular differential ideals without any use of Gröbner bases. This algorithm simplifies the theory (somehow a “pedagogic” contribution) but permits us also to perform easily linear algebra over the base field in the factor differential ring defined by a regular differential ideal.