HILBERT-SAMUEL FUNCTION OF CERTAIN GRADED ALGEBRAS

Let k be a field and A=(A(n))n greater-than-or-equal-to 1 a graded k-algebra with N degree 1 operators. Assume A is generated by its degree 1 elements as an algebra with operators. Under a weak associativity condition, we prove the inequality, for all n, r greater-than-or-equal-to 1: dim A(n+r) greater-than-or-equal-to (2 n dim A(n) + N)r dim A(n). In particular, dim A(n+1) is bounded independently of dim A(i) for i<n. This applies notably to associative algebras. Lie algebras and to the associated graded of certain kinds of descending central series on a group.