AAECC 2, 21-33 (1991) AAECC ApplicableAlgebra in Engineering, Communication end Computing © Springer-Verlag 1991 On the Computation of Hilbert-Poincar6 Series* Anna Maria Bigatti, Massimo Caboara and Lorenzo Robbiano** Departimento di Matematica dell'Universitfi di Genova, Via L. B. Alberti 4, I- 16132 Genova Italy Received November 5, 1990 Abstract. We prove a theorem, which provides a formula for the computation of the Poincar6 series ofa monomial ideal in k[X 1 ..... Xn].via the computation of the Poincar6 series of some monomial ideals in k[X 1 ..... X i..... X,]. The complexity of our algorithm is optimal for Borel-normed ideals and an implementation in CoCoA strongly confirms its efficiency. An easy extension computes the Poincar6 series of graded modules over standard algebras. Keywords: Hilbert functions, Poincar6 series, Borel-normed ideals Introduction There is no doubt that Hilbert functions of standard algebras are a fundamental tool in Commutative Algebra and Algebraic Geometry. If X is a projective variety or scheme in IP~, (k a field) and A its projective coordinate ring, then the number Hx(d):=dimkAd was classically called the postulation of X in degree d. It means the number of independent linear conditions that are imposed to the coefficients of a generic hypersurface of degree d in order to contain X. The function Hx is nowadays called the Hilbert function of X and more generally if R:=-k[X1 ..... X.] (k a field) is graded by deg(X~)= 1, i= 1..... n and I is a homogeneous ideal of R, then R/I is called a standard k-algebra (see Stanley (1978)) and the function HR/1 defined by HR/I(n):= dim k (R/1), is called the Hilbert function of R/I. It is well-known that for n >>0, HR/I(n ) takes the same values as a polynomial PR/I, called the Hilbert polynomial of R/I. But Hilbert functions contain more * E-mail: robbiano@ igecuniv.bitnet ** The paper was partly written while the third author was visiting Queen's University, during the academic year 1989/90. It was partly supported by the Natural Sciences & Engineering Research Council of Canada, Queen's University (Kingston, Canada) and Consiglio Nazionale delle Ricerche