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

- 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

## Description

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

## Ways to use this method:

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

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/functions/hilbertSeries-doc.m2:46:0.