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 which 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, MonomialOrder => RevLex, Inverses => true]
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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 3 2 4 3 5 4 6 4 7
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 , MonomialOrder => RevLex, Inverses => true]
0 1
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