hilbertSeries(Module) -- compute the Hilbert series of the module

Synopsis

• Function: hilbertSeries
• Usage:
hilbertSeries M
• Inputs:
• M,
• 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 a module.
 i1 : R = ZZ/101[x, Degrees => {2}]; i2 : M = module ideal x^2 o2 = image | x2 | 1 o2 : R-module, submodule of R i3 : s = hilbertSeries M 4 T o3 = -------- 2 (1 - T ) o3 : Expression of class Divide i4 : numerator s 4 o4 = T o4 : ZZ[T] i5 : poincare M 4 o5 = T o5 : ZZ[T]
Recall that the variables of the power series are the variables of the degrees ring.
 i6 : R=ZZ/101[x, Degrees => {{1,1}}]; i7 : M = module ideal x^2; i8 : s = hilbertSeries M 2 2 T T 0 1 o8 = ---------- (1 - T T ) 0 1 o8 : Expression of class Divide i9 : numerator s 2 2 o9 = T T 0 1 o9 : ZZ[T ..T ] 0 1 i10 : poincare M 2 2 o10 = T T 0 1 o10 : ZZ[T ..T ] 0 1