  We prove an effective bound for the degrees of generators of the algebraic de
Rham cohomology of smooth affine hypersurfaces. In particular, we show that the
de Rham cohomology H_dR^p(X) of a smooth hypersurface X of degree d in C^n can
be generated by differential forms of degree d^O(pn). 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.
