On the computation of Hilbert-Poincare` Series

We prove a theorem, which provides a formula for the computation of the Poincar\'e series of a monomial ideal in $k[X_1,\dots,X_n]$, via the computation of the Poincare' series of some monomial ideals in $k[X_1,\dots ,\widehat{X_i}, \dots,X_n]$. 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 Poincare' series of graded modules over standard algebras.