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];
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i2 : hilbertSeries(R/x^3)
3
1 - T
o2 = --------
2
(1 - T)
o2 : Expression of class Divide
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i3 : hilbertSeries(R/x^3, Order =>5)
2 3 4
o3 = 1 + 2T + 3T + 3T + 3T
o3 : ZZ[T]
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If the ambient ring is multigraded, then the
degrees ring has multiple variables.
i4 : R = ZZ/101[x,y, Degrees=>{{1,2},{2,3}}];
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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
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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
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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
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i8 : degrees ring oo
o8 = {{-1}, {1}}
o8 : List
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i9 : heft R
o9 = {-1, 1}
o9 : List
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