dehn-sommerville relations, upper bound theorem, and levels in arrangements
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 l th 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.
arrangement general invariance functional vectorization