Unmixed-dimensional Decomposition of a Finitely Generated Perfect Differential Ideal

Abstract

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.

Computing representations for radicals of finitely generated differential ideals

“…Note that the membership problem in an arbitrary differential ideal is undecidable [Gallo et al, 1991], and the membership problem of a finitely generated differential ideal is still open. Bouziane et al [2001] and Maârouf [1996] designed recently a variant of the Rosenfeld-Gröbner algorithm. They started from the algorithm of Kalkbrener [1993] which compute decompositions of radicals of ideals in non differential polynomial algebras.…”

On the Regularity Property of Differential Polynomials Modulo Regular Differential Chains

“…Another algorithm for computing the inverse of a polynomial modulo a regular chain can be found in [6,Algorithm Invert]. It is based on a Gröbner basis computation.…”

“…modulo the radical of the ideal defined by the regular chain. However, [6] misses [5, Theorem 1.6] which implies that B has the same set of associated prime ideals as its radical, hence that the computed inverse also is an inverse modulo B.…”

“…However, [18] misses [5,Theorem 1.6] which states that this property is held by all the associated prime ideals of B. See the comments on [6,Algorithm Invert].…”

Ritt-Wu's characteristic set method for ordinary difference polynomial systems with arbitrary ordering

“…The idea of the method is to decompose the zero set of an equation system into the union of the zero sets of equation systems in triangular forms. The characteristic set method has been developed for polynomial systems [1,5,15,19] and differential polynomial systems [2,3,9,10,13,16].…”