Resolution
Global GMS
ES Strata
Build. Blocks
Comb. Appl.
HCA Proving
Arrangements
Branches
Classify
Coding
Deformations
Equidim Part
Existence
Finite Groups
Flatness
Genus
Hilbert Series
Membership
Nonnormal Locus
Normalization
Primdec
Puiseux
Plane Curves
Saturation
Solving
Space Curves
Spectrum
Hilbert Series
Let M be a graded module over K[x1,...,xn] with respect to weights w1,...,wn .

The Hilbert function HM is defined (on Z) by HM(k) := dimKMk .

The Hilbert-Poincaré series of M is the power series
$\displaystyle HP_M(t) = \sum_{i=
It turns out that HPM(t) can be written in two useful ways for weights (1,...,1) :
$\displaystyle HP_M(t) =
 \frac{Q(t)}{(1-t)^n}=\frac{P(t)}{(1-t)^{\dim(M)}}$


where Q(t) and P(t) are polynomials with coefficients in Z.
  • Q(t) is called the first Hilbert series,
  • P(t) is called the second Hilbert series.
For general w1,...,wn :
$\displaystyle HP_M(t) = \frac{Q(t)}{\prod\limits_{i=

where Q(t) is a polynomial in Z[t], the first (weighted) Hilbert series of M . An Example

KL, 06/03 http://www.singular.uni-kl.de