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Macaulay2Doc :: hilbertSeries(..., Order => ...)

hilbertSeries(..., Order => ...) -- display the truncated power series expansion

Synopsis

Description

We compute the Hilbert series both without and with the optional argument. In the second case notice the terms of power series expansion up to, but not including, degree 5 are displayed rather than expressing the series as a rational function. The polynomial expression is an element of a Laurent polynomial ring that is the degrees ring of the ambient ring.
i1 : R = ZZ/101[x,y];
i2 : hilbertSeries(R/x^3)

           3
      1 - T
o2 = --------
            2
     (1 - T)

o2 : Expression of class Divide
i3 : hilbertSeries(R/x^3, Order =>5)

                2     3     4
o3 = 1 + 2T + 3T  + 3T  + 3T

o3 : ZZ[T]
If the ambient ring is multigraded, then the degrees ring has multiple variables.
i4 : R = ZZ/101[x,y, Degrees=>{{1,2},{2,3}}];
i5 : hilbertSeries(R/x^3, Order =>5)

            2    2 4    2 3    3 5    4 7    4 6
o5 = 1 + T T  + T T  + T T  + T T  + T T  + T T
          0 1    0 1    0 1    0 1    0 1    0 1

o5 : ZZ[T , T ]
         0   1
The heft vector provides a suitable monomial ordering and degrees in the ring of the Hilbert series.
i6 : R = QQ[a..d,Degrees=>{{-2,-1},{-1,0},{0,1},{1,2}}]

o6 = R

o6 : PolynomialRing
i7 : hilbertSeries(R, Order =>3)

            2         -1    -2 -1    2 4      3     2     -1       -2  
o7 = 1 + T T  + T  + T   + T  T   + T T  + T T  + 2T  + 2T  T  + 2T   +
          0 1    1    0     0  1     0 1    0 1     1     0  1     0   
     ------------------------------------------------------------------------
      -3 -1    -4 -2
     T  T   + T  T
      0  1     0  1

o7 : ZZ[T , T ]
         0   1
i8 : degrees ring oo

o8 = {{-1}, {1}}

o8 : List
i9 : heft R

o9 = {-1, 1}

o9 : List

Further information