Dehn-Sommerville relations, upper bound theorem, and levels in arrangements

Abstract

In this note, we generalize the h-vector for simple, bounded convex polytopes [14] to the h-matrix for simple, bounded k-complexes. We observe that the h-matrix is invariant with respect to the defining linear function, and that the Dehn-Sommerville relations and McMullen's Upper Bound Theorem [13] for convex polytopes follow from the invariance ofthe 0-th row and column of this matrix. The invariance of the other entries in the h-matrix should, perhaps, be investigated more. One new consequence is that, given any non-degenerate linear function z, the number of local z-minima on the lth level of any d-dimensional arrangement is bounded by (l+d-1 / d-1) with exact equality if the l-th level is bounded and simple.