  We present a new upper bound for the orders of derivatives in the
Rosenfeld-Groebner algorithm. This algorithm computes a regular decomposition
of a radical differential ideal in the ring of differential polynomials over a
differential field of characteristic zero with an arbitrary number of commuting
derivations. This decomposition can then be used to test for membership in the
given radical differential ideal. In particular, this algorithm allows us to
determine whether a system of polynomial PDEs is consistent. Previously, the
only known order upper bound was given by Golubitsky, Kondratieva, Moreno Maza,
and Ovchinnikov for the case of a single derivation. We achieve our bound by
associating to the algorithm antichain sequences whose lengths can be bounded
using the results of Leon Sanchez and Ovchinnikov.
