  In this paper we study the problem of computing a Kolchin characteristic set
of a radical differential ideal. The central part of the article is the
presentation of algorithms solving this problem in two principal cases: for
ordinary differential polynomials and in the partial differential case. Our
computations are mainly performed with respect to orderly rankings. We also
discuss the usefulness of regular and characteristic decompositions of radical
differential ideals. In the partial differential case we give an algorithm for
computing characteristic sets in the special case of radical differential
ideals satisfying the property of consistency. For this class of ideals we show
how to deal with arbitrary differential rankings.
