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

Synopsis

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

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, MonomialOrder => RevLex, Inverses => true]

i4 : numerator reduceHilbert hilbertSeries I

                2    3
o4 = 1 + 2T + 2T  - 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 : I = ideal x^2;

o6 : Ideal of R

i7 : poincare I

          2 2
o7 = 1 - T T
          0 1

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

i8 : numerator reduceHilbert hilbertSeries I

o8 = 1 + T T
          0 1

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

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.