  The notion of differentiation index for DAE systems of arbitrary order with
generic second members is discussed by means of the study of the behavior of
the ranks of certain Jacobian associated sub-matrices. As a by-product, we
obtain upper bounds for the regularity of the Hilbert-Kolchin function and the
order of the ideal associated to the DAE systems under consideration, not
depending on characteristic sets. Some quantitative and algorithmic results
concerning differential transcendence bases and induced equivalent explicit ODE
systems are also established.
