hilbertSeries -- compute Hilbert series

hilbertSeries -- compute Hilbert series

hilbertSeries M -- compute the Hilbert series of the ring or module M.

The Hilbert series is the formal power series in the variables of the degrees ring whose coefficients are the dimensions of the corresponding graded component. The series is provided as an Expression representing a rational function with that series.

If an optional integer argument labelled Order is used, then the power series is expanded to that order.

i1 : R = ZZ/101[x, Degrees => {2}];

i2 : hilbertSeries(R/x^2)



            4

      1 - $T

o2 = ---------

            2

     (1 - $T )



o2 : Divide

i3 : numerator oo



           4

o3 = 1 - $T



o3 : ZZ[ZZ^1]

i4 : value oo



           4

o4 = 1 - $T



o4 : ZZ[ZZ^1]

i5 : poincare (R/x^2)



           4

o5 = 1 - $T



o5 : ZZ[ZZ^1]

i6 : hilbertSeries(R/x^2, Order => 12)



           2

o6 = 1 + $T



o6 : ZZ[ZZ^1]

i7 : R=ZZ/101[x, Degrees => {{1,1}}];

i8 : hilbertSeries (R/x^2)



            2  2

      1 - $T $T

            0  1

o8 = ------------

     (1 - $T $T )

            0  1



o8 : Divide

See also:

degreesRing -- the ring of degrees

Order -- specify the order of a Hilbert series required

Ways to use hilbertSeries :

hilbertSeries CoherentSheaf

hilbertSeries Ideal

hilbertSeries Module

hilbertSeries PolynomialRing

hilbertSeries ProjectiveHilbertPolynomial

hilbertSeries ProjectiveVariety

hilbertSeries QuotientRing

Optional arguments :

hilbertSeries(..., Order) -- specify the order of a Hilbert series required