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poincare(Arrangement) -- compute the Poincaré polynomial of an arrangement



The Poincaré polynomial $\pi({\mathcal A},t)$ of a central arrangement of rank $r$ equals $t^r\,T(1+t^{-1},0)$, where $T(x,y)$ is the Tutte polynomial. Alternatively, \[ \pi({\mathcal A},t)=\sum_F\mu(\widehat{0},F)(-t)^{r(F)}, \] where the sum is over all flats $F$, the function $\mu$ denotes the Möbius function of the intersection lattice, and $r(F)$ is the rank of the flat $F$. The characteristic polynomial of an (essential) arrangement is closely related and defined by \[ \chi({\mathcal A},t)=t^r\pi({\mathcal A},-t^{-1}). \]

i1 : A = arrangement "MacLane";
i2 : poincare A

                 2      3
o2 = 1 + 8T + 20T  + 13T

o2 : ZZ[T]
i3 : characteristicPolynomial matroid A

      3     2
o3 = x  - 8x  + 20x - 13

o3 : ZZ[x]

If ${\mathcal A}$ is an arrangement defined over the complex numbers, a classical theorem of Brieskorn-Orlik-Solomon asserts that $\pi({\mathcal A},t)$ is also the Poincaré polynomial of the complement of the union of hyperplanes. In certain interesting cases, the Poincaré polynomial factors into linear factors. This is the case if ${\mathcal A}$ is the set of reflecting hyperplanes associated with a real or complex reflection group, in which case the (co)exponents of the reflection group appear as the linear coefficients of the factors.

i4 : factor poincare typeA 3

o4 = (1 + T)(1 + 2T)(1 + 3T)

o4 : Expression of class Product

More generally (since reflection arrangements are free), if the module of logarithmic derivations $D({\mathcal A})$ on $\mathcal A$ is free, Terao's Factorization Theorem states that the Poincaré polynomial factors as a product $\prod_{i=1}^r(1+m_i t)$, where the $m_i$'s are the degrees of the generators of the graded free module $D({\mathcal A})$.

i5 : A = arrangement "Hessian";
i6 : factor poincare A

o6 = (1 + T)(1 + 4T)(1 + 7T)

o6 : Expression of class Product
i7 : prune image der A

        ZZ          3
o7 = (-----[x ..x ])
      31627  1   3

o7 : -----[x ..x ]-module, free, degrees {1, 4, 7}
     31627  1   3

The Poincaré polynomial appears in various enumerative contexts as well. If ${\mathcal A}$ is an arrangement defined over the real numbers, then $\pi({\mathcal A},1)$ equals the number of connected components in the complement of the union of hyperplanes. Similarly, $d/dt[\pi({\mathcal A},t)]$ evaluated at $t=1$ counts the number of bounded components in the complement of the decone of ${\mathcal A}$.

If ${\mathcal A}$ is a non-central arrangement, the Poincaré polynomial $\pi({\mathcal A},t)$ equals $\pi(c{\mathcal A},t)/(1+t)$, where $c{\mathcal A}$ denotes the cone of ${\mathcal A}$.

See also

Ways to use this method: