Effective de Rham cohomology

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 HpdR(X) of a smooth hypersurface X of degree d in Cn can be generated by differential forms of degree dO(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.