  Elimination of unknowns in systems of equations, starting with Gaussian
elimination, is a problem of general interest. The problem of finding an a
priori upper bound for the number of differentiations in elimination of
unknowns in a system of differential-algebraic equations (DAEs) is an important
challenge, going back to Ritt (1932). The first characterization of this via an
asymptotic analysis is due to Grigoriev's result (1989) on quantifier
elimination in differential fields, but the challenge still remained.
  In this paper, we present a new bound, which is a major improvement over the
previously known results. We also present a new lower bound, which shows
asymptotic tightness of our upper bound in low dimensions, which are frequently
occurring in applications. Finally, we discuss applications of our results to
designing new algorithms for elimination of unknowns in systems of DAEs.
