Editing Arrangement of hyperplanes (section)

=== Polynomials ===

For a subset ''B'' of ''A'', let us define ''f''(''B'') := the intersection of the hyperplanes in ''B''; this is ''S'' if ''B'' is empty.  
The '''characteristic polynomial of''' ''A'', written ''p<sub>A</sub>''(''y''), can be defined by 

:<math>p_A(y) := \sum_B (-1)^{|B|}y^{\dim f(B)},</math>

summed over all subsets ''B'' of ''A'' except, in the affine case, subsets whose intersection is empty.  (The dimension of the empty set is defined to be &minus;1.)  This polynomial helps to solve some basic questions; see below.
Another polynomial associated with ''A'' is the '''Whitney-number polynomial''' ''w<sub>A</sub>''(''x'', ''y''), defined by

:<math>w_A(x,y) := \sum_B x^{n-\dim f(B)} \sum_C (-1)^{|C-B|}y^{\dim f(C)},</math>

summed over ''B'' ⊆ ''C'' ⊆ ''A'' such that ''f''(''B'') is nonempty.

Being a geometric lattice or semilattice, ''L''(''A'') has a characteristic polynomial, ''p''<sub>''L''(''A'')</sub>(''y''), which has an extensive theory (see [[Matroid#Characteristic_polynomial|matroid]]).  Thus it is good to know that ''p''<sub>''A''</sub>(''y'') = ''y''<sup>''i''</sup> ''p''<sub>''L''(''A'')</sub>(''y''), where ''i'' is the smallest dimension of any flat, except that in the projective case it equals ''y''<sup>''i'' + 1</sup>''p''<sub>''L''(''A'')</sub>(''y'').  
The Whitney-number polynomial of ''A'' is similarly related to that of ''L''(''A'').  
(The empty set is excluded from the semilattice in the affine case specifically so that these relationships will be valid.)