hilbertSeries

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)



        3     2

     ($T  + $T  + $T + 1)(- $T + 1)

o2 = ------------------------------

                    2

               (- $T  + 1)



o2 : Divide

i3 : numerator oo



        3     2

o3 = ($T  + $T  + $T + 1)(- $T + 1)



o3 : Product

i4 : value oo



         4

o4 = - $T  + 1



o4 : ZZ[ZZ^1]

i5 : poincare (R/x^2)



         4

o5 = - $T  + 1



o5 : ZZ[ZZ^1]

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



       2

o6 = $T  + 1



o6 : ZZ[ZZ^1]

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

i8 : hilbertSeries (R/x^2)



          2  2

      - $T $T  + 1

          0  1

o8 = --------------

     (- $T $T  + 1)

          0  1



o8 : Divide