hilbertSeries(Ideal) -- compute the Hilbert series of the quotient of the ambient ring by the ideal

Synopsis

• Function: hilbertSeries
• Usage:
hilbertSeries I
• Inputs:
• Optional inputs:
• Order => ..., default value infinity, display the truncated power series expansion
• Reduce => ..., default value false, reduce the Hilbert series
• Outputs:
• , the Hilbert series

Description

We compute the Hilbert series of R/I, the quotient of the ambient ring by the ideal. Caution: For an ideal I running hilbertSeries I calculates the Hilbert series of R/I.
 i1 : R = ZZ/101[x, Degrees => {2}]; i2 : I = ideal x^2 2 o2 = ideal x o2 : Ideal of R i3 : s = hilbertSeries I 4 1 - T o3 = -------- 2 (1 - T ) o3 : Expression of class Divide i4 : numerator s 4 o4 = 1 - T o4 : ZZ[T] i5 : poincare I 4 o5 = 1 - T o5 : ZZ[T] i6 : reduceHilbert s 2 1 + T o6 = ------ 1 o6 : Expression of class Divide
Recall that the variables of the power series are the variables of the degrees ring.
 i7 : R=ZZ/101[x, Degrees => {{1,1}}]; i8 : I = ideal x^2; o8 : Ideal of R i9 : s = hilbertSeries I 2 2 1 - T T 0 1 o9 = ---------- (1 - T T ) 0 1 o9 : Expression of class Divide i10 : numerator s 2 2 o10 = 1 - T T 0 1 o10 : ZZ[T ..T ] 0 1 i11 : poincare I 2 2 o11 = 1 - T T 0 1 o11 : ZZ[T ..T ] 0 1 i12 : reduceHilbert s 1 + T T 0 1 o12 = -------- 1 o12 : Expression of class Divide

Caveat

As is often the case, calling this function on an ideal I actually computes it for R/I where R is the ring of I.

Ways to use this method:

• hilbertSeries(Ideal) -- compute the Hilbert series of the quotient of the ambient ring by the ideal