  We consider the Rosenfeld-Groebner algorithm for computing a regular
decomposition of a radical differential ideal generated by a set of ordinary
differential polynomials in n indeterminates. For a set of ordinary
differential polynomials F, let M(F) be the sum of maximal orders of
differential indeterminates occurring in F. We propose a modification of the
Rosenfeld-Groebner algorithm, in which for every intermediate polynomial system
F, the bound M(F) is less than or equal to (n-1)!M(G), where G is the initial
set of generators of the radical ideal. In particular, the resulting regular
systems satisfy the bound. Since regular ideals can be decomposed into
characterizable components algebraically, the bound also holds for the orders
of derivatives occurring in a characteristic decomposition of a radical
differential ideal.
  We also give an algorithm for converting a characteristic decomposition of a
radical differential ideal from one ranking into another. This algorithm
performs all differentiations in the beginning and then uses a purely algebraic
decomposition algorithm.
