Unmixed-dimensional Decomposition of a Finitely Generated Perfect Differential Ideal

We present an algorithm that computes an unmixed-dimensional decomposition of a finitely generated perfect differential ideal I. Each Iiin the decompositionI = I1 ∩ ∩ Ikis given by its characteristic set. This decomposition is a generalization of the differential case of Kalkbrener’s decomposition. We use a different approach. The basic operation in our algorithm is the computation of the inverse of an algebraic polynomial with respect to a finite set of algebraic polynomials. No factorization is needed. Some of the main problems in polynomial ideal theory can be solved by means of this decomposition: we show how the radical membership can be decided, a characteristic set of a prime differential ideal can be selected, and the differential dimension with a parametric set of a differential ideal can be read. The algorithm has been implemented in the computer algebra system MAPLE and has been tested successfully on many examples.