  We study a generalization of a conjecture made by Beauville on the Chow ring
of hyper-K\"ahler algebraic varieties. Namely we prove in a number of cases
that polynomial cohomological relations involving only CH^1(X) and the Chern
classes of X are satisfied in CH(X). These cases are : punctual Hilbert schemes
of a K3 surface S parameterizing subschemes of length n, for n<2b_2(S)_tr+5;
Fano varieties of lines in a cubic fourfold.
