hilbertSeries(PolynomialRing) -- compute the Hilbert series of a ring

Synopsis

Function: hilbertSeries
Usage:

hilbertSeries R
Inputs:
- R, a polynomial ring
Optional inputs:
- Order => ..., default value infinity, display the truncated power series expansion
- Reduce => ..., default value false, reduce the Hilbert series
Outputs:
- a divide expression, the Hilbert series

We compute the Hilbert series of a ring.

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

i2 : s = hilbertSeries R

           4
      1 - T
o2 = --------
           2
     (1 - T )

o2 : Expression of class Divide

i3 : numerator s

          4
o3 = 1 - T

o3 : ZZ[T]

i4 : poincare R

          4
o4 = 1 - T

o4 : ZZ[T]

Recall that the variables of the power series are the variables of the degrees ring.

i5 : R=ZZ/101[x, Degrees => {{1,1}}]/ideal(x^2);

i6 : s = hilbertSeries R

           2 2
      1 - T T
           0 1
o6 = ----------
     (1 - T T )
           0 1

o6 : Expression of class Divide

i7 : numerator s

          2 2
o7 = 1 - T T
          0 1

o7 : ZZ[T ..T ]
         0   1

i8 : poincare R

          2 2
o8 = 1 - T T
          0 1

o8 : ZZ[T ..T ]
         0   1