poincare(Module) -- assemble degrees of an module into a polynomial

Synopsis

Usage:
poincare M
Function: poincare
Inputs:
- M, a module
Outputs:
- a ring element, in the Laurent polynomial ring whose variables correspond to the degrees of the ambient ring

Description

We compute the Poincare polynomial of a module.

i1 : R = ZZ/101[w..z];

i2 : M = module monomialCurveIdeal(R,{1,3,4});

i3 : poincare M

      2     3     4    5
o3 = T  + 3T  - 4T  + T

o3 : ZZ [T, MonomialOrder => RevLex, Inverses => true]

i4 : numerator reduceHilbert hilbertSeries M

      2     3     4    5
o4 = T  + 3T  - 4T  + T

o4 : ZZ [T, MonomialOrder => RevLex, Inverses => true]

Recall that the variables of the polynomial are the variables of the degrees ring.

i5 : R=ZZ/101[x, Degrees => {{1,1}}];

i6 : M = module ideal x^2;

i7 : poincare M

      2 2
o7 = T T
      0 1

o7 : ZZ [T , T , MonomialOrder => RevLex, Inverses => true]
          0   1

i8 : numerator reduceHilbert hilbertSeries M

      2 2
o8 = T T
      0 1

o8 : ZZ [T , T , MonomialOrder => RevLex, Inverses => true]
          0   1