Estimates for the coefficients of differential dimension polynomials

We answer the following long-standing question of Kolchin: given
a system of algebraic-dierential equations (x1; : : : ; xn) = 0 in m derivatives
over a dierential eld of characteristic zero, is there a computable bound,
that only depends on the order of the system (and on the xed data m and n),
for the typical dierential dimension of any prime component of ? We give
a positive answer in a strong form; that is, we compute a (lower and upper)
bound for all the coecients of the Kolchin polynomial of every such prime
component. We then show that, if we look at those components of a specied
dierential type, we can compute a signicantly better bound for the typical
dierential dimension. This latter improvement comes from new combinatorial
results on characteristic sets, in combination with the classical theorems of
Macaulay and Gotzmann on the growth of Hilbert-Samuel functions.