 The Hilbert function
maps the integer onto the dimension of the -vector space .The Hilbert series, which is called Hilbert–Poincaré series in the more general setting of graded vector spaces, is the formal series

If is generated by homogeneous elements of positive degrees formula_7, then the sum of the Hilbert series is a rational fraction
where is a polynomial with integer coefficients.If is generated by elements of degree 1 then the sum of the Hilbert series may be rewritten as
where is a polynomial with integer coefficients, and formula_10 is the Krull dimension of .