poincare(Ring) -- assemble degrees of an ring into a polynomial

Synopsis

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

We compute the Poincare polynomial of a ring.

i1 : R=ZZ/101[x]/ideal(x^2);

i2 : poincare R

          2
o2 = 1 - T

o2 : ZZ [T, MonomialOrder => RevLex, Inverses => true]

i3 : numerator hilbertSeries R

          2
o3 = 1 - T

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

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

i4 : R=ZZ/101[x, Degrees => {{1,1}}]/ideal(x^2);

i5 : poincare R

          2 2
o5 = 1 - T T
          0 1

o5 : ZZ [T , T , MonomialOrder => RevLex, Inverses => true]
          0   1

i6 : numerator hilbertSeries R

          2 2
o6 = 1 - T T
          0 1

o6 : ZZ [T , T , MonomialOrder => RevLex, Inverses => true]
          0   1