  We answer the following long-standing question of Kolchin: given a system of
algebraic-differential equations $\Sigma(x_1,\dots,x_n)=0$ in $m$ derivatives
over a differential field of characteristic zero, is there a computable bound,
that only depends on the order of the system (and on the fixed data $m$ and
$n$), for the typical differential dimension of any prime component of
$\Sigma$? We give a positive answer in a strong form; that is, we compute a
(lower and upper) bound for all the coefficients of the Kolchin polynomial of
every such prime component. We then show that, if we look at those components
of a specified differential type, we can compute a significantly better bound
for the typical differential 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.
