We present an algorithm that computes an unmixed-dimensional decomposition of a finitely generated perfect differential ideal I. Each I i in the decomposition I = I 1 ∩ · · · ∩ I k is 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.
We give an algorithm to compute a resolvent of an algebraic variety without computing its irreducible components; we decompose the radical of an ideal into prime ideals and we test the primality of a regular ideal.
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