  Grothendieck has proved that each class in the de Rham cohomology of a smooth
complex affine variety can be represented by a differential form with
polynomial coefficients. After having proved a single exponential bound for the
degrees of these forms in the case of a hypersurface, here we generalize this
result to arbitrary codimension. More precisely, we show that the p-th de Rham
cohomology of a smooth affine variety of dimension m and degree D can be
represented by differential forms of degree (pD)^{O(pm)}. This result is
relevant for the algorithmic computation of the cohomology, but is also
motivated by questions in the theory of ordinary differential equations related
to the infinitesimal Hilbert 16th problem.
