Arrangements in unitary and orthogonal geometry over finite fields (Q1067206)

Let V be an n-dimensional vector space over \({\mathbb{F}}_ q\) and \(\phi\) a Hermitian form in V with respect to an automorphism \(\sigma\) with \(\sigma^ 2=1\); here \(\sigma =1\) is allowed, but only in case q is odd. L is the geometric lattice consisting of the intersections of non- isotropic hyperplanes (ordered by reverse inclusion), with characteristic polynomial \(\chi (L,t)=\sum_{X\in L}\mu (V,X)t^{\dim X},\) where \(\mu\) denotes the Möbius function. It is shown in this paper that \(\chi (L,t)=(t-1)(t-q)(t-q^ 2)...(t-q^{n-\nu -1})\gamma (t),\) and that the monic polynomial \(\gamma\) (t)\(\in {\mathbb{Z}}[t]\) of degree \(\nu\) \((= the\) Witt index of \(\phi)\) has in general no integer roots if \(\nu\geq 2\).