# 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:
• , 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.