On the computation of the Hilbert series

Abstract

We present an algorithm to compute the Hilbert series of a homogeneous ideal in a polynomial ring. From an ideal I, generated by the principal monomials of a Gröbner basis, we compute the Hilbert series $Hilb_{k[x_1 ,...,x_n ]/I}$(t)=T _I (t)/(1−t)ⁿ. The polynomial T _I is computed by recursively factoring out a monomial m of degree d at each step

$$T_I (t) = T_{I + (m)} (t) + t^d T_{I + Ann(m)} (t).$$

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Supported by the Swedish National Board for Technical Development.

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