The problem of efficiently transforming differential sets from one ranking to another is studied in this paper. The major advantage of the authors’ approach over other approaches is the fact that the most time-consuming part of the developed algorithm is handled by applying pure algebraic operations.

The main results are heavily based on the use of the concept of bound, which is studied in three steps. During the first step, it is proved that there exists a characteristic set for any differential ideal and for an arbitrary ranking, so that the orders of its elements are bounded by the order of the ideal. During the second step, the above statement about existence is generalized for the case of characterizable ideals. In the third step, it is shown that the bound actually holds for canonical characteristic sets of characterizable differential ideals. The bound is used to determine how many times one should differentiate the differential polynomials, so that the characteristic decomposition can be computed by using only algebraic operations.

Some introductory remarks are given in the first section. Section 2 introduces the necessary differential algebraic notation. The algebraic algorithm for converting characteristic decompositions from one ranking to another is developed in sections 3 and 4. Canonical characteristic sets are discussed in section 5. Section 6 prepares the proof of the bound, and section 7 shows how to compute the canonical sets from any other known representation of a characterizable differential ideal.