Triangular matrices, differential resultants and systems of linear homogeneous PDE's

In this paper, it is shown that if we consider an upper triangular form T(RM) of RM using only row operations, then the differential resultants with respect to a permutation σ can be defined as divisors of the entries of T(RM) in the last column and in the rows with only a nonzero entry. Furthermore, a triangular form for the system is provided. Unfortunately the corresponding system is not always completely integrable, on the other hand such triangular form is useful for finding the integrability conditions.