# poincare(ChainComplex) -- assemble degrees of a chain complex into a polynomial

## Synopsis

• Function: poincare
• Usage:
poincare C
• Inputs:
• C, ,
• Outputs:
• , in the Laurent polynomial ring degrees ring, whose variables correspond to the degrees of the ambient ring

## Description

We compute poincare for a chain complex.

 i1 : R = ZZ/32003[a..h]; i2 : C = res ideal(a*b, c*d, e*f) 1 3 3 1 o2 = R <-- R <-- R <-- R <-- 0 0 1 2 3 4 o2 : ChainComplex i3 : poincare C 2 4 6 o3 = 1 - 3T + 3T - T o3 : ZZ[T]

Note that since the Hilbert series is additive in exact sequences, for a free resolution this only depends on the Betti numbers of the resolution. For more details, see Hilbert functions and free resolutions.

 i4 : b = betti C 0 1 2 3 o4 = total: 1 3 3 1 0: 1 . . . 1: . 3 . . 2: . . 3 . 3: . . . 1 o4 : BettiTally i5 : poincare b 2 4 6 o5 = 1 - 3T + 3T - T o5 : ZZ[T]