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

## Synopsis

• Usage:
hilbertSeries(..., Order => n)
• Inputs:
• Consequences:

## 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

• Default value: infinity
• Function: hilbertSeries -- compute the Hilbert series
• Option key: Order -- specify the order of a Hilbert series required