Hilbert polynomials and powers of ideals

The growth of Hilbert coefficients for powers of ideals are Studied. For a graded ideal I in the polynomial ring S = K[x(1),.... x(n)] and a finitely generated graded S-module M, the Hilbert coefficients e(i)(M/I(k)M) are polynomial functions. Given two families of graded ideals (I(k))(k >= 0) and (J(k))(k >= 0) with J(k) subset of I(k) for all k with the property that J(k)K(l) subset of J(k+l) and I(k)I(l) subset of I(k+l) for all k and l, and Such that the algebras A = circle plus(k >= 0) J(k) and B = circle plus(k >= 0) I(k) are finitely generated, we show the function k |-> e(0)(I(k)/J(k)) is of quasi-polynomial type, say given by the polynomials P(0),...,P(g-1). If J(k) = J(k) for all k, for a graded ideal J, then we show that all the Pi have the same degree and the same leading coefficient. As one of the applications it is shown that lim(k ->infinity) l(Gamma(m)(S/I(k)))/k(n) is an element of Q, if I is a monomial ideal. We also Study analogous statements in the local case