An upper bound theorem for rational polytopes (Q1268610)

This nice note presents a generalization of the well-known inequality \[ h_i(P)-h_{i-1}(P) \leq {n-d+i-2 \choose i}, \quad (0 \leq i \leq d/2) \] in the case of the boundary of simplicial polytopes to the case of rational polytopes (those polytopes that have a coordinatization with rational coordinates for the vertices). In that case \(h(P)\) denotes the so called toric \(h\)-vector [for a short overview on the topic see \textit{L. Billera} and \textit{A. Björner}, in: Handbook of discrete and computational geometry. 291-310 (1997; see the paper above)]. The proof relies on previous work by the author and work by Braden and MacPherson that implies that the \(g\)-polynomial of any quotient of a rational polytope is bounded by the \(g\)-polynomial of the polytope.