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.
Abstract. The singularity problem of the solutions of some particular Sylvester equations is studied. As a consequence of this study, a good choice of a Sylvester equation which is associated to a linear continuous time system can be made such that its solution is nonsingular. This solution is then used to solve an eigenstructure assignment problem for this system. From a practical point view, this study can also be applied to automatic control when the system is subject to input constraints.
In this paper, polynomial ideal theory is used to deal with the problem of the $S$-packing coloring of a finite undirected and unweighted graph by introducing a family of polynomials encoding the problem. A method to find the $S$-packing colorings of the graph is presented and illustrated by examples.
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