poincare(Ideal)  assemble degrees of the quotient of the ambient ring by an ideal into a polynomial
Synopsis

Function: poincare

 Usage:
 poincare I

Inputs:

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 the quotient of the ambient ring by an ideal.
i1 : R = ZZ/101[w..z];

i2 : I = monomialCurveIdeal(R,{1,3,4});
o2 : Ideal of R

i3 : poincare I
2 3 4 5
o3 = 1  T  3T + 4T  T
o3 : ZZ[T]

i4 : numerator reduceHilbert hilbertSeries I
2 3
o4 = 1 + 2T + 2T  T
o4 : ZZ[T]

Recall that the variables of the polynomial are the variables of the degrees ring.
i5 : R=ZZ/101[x, Degrees => {{1,1}}];

i6 : I = ideal x^2;
o6 : Ideal of R

i7 : poincare I
2 2
o7 = 1  T T
0 1
o7 : ZZ[T ..T ]
0 1

i8 : numerator reduceHilbert hilbertSeries I
o8 = 1 + T T
0 1
o8 : ZZ[T ..T ]
0 1

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.