an:03988122
Zbl 0611.52002
Stanley, Richard P.
On the number of faces of centrally-symmetric simplicial polytopes
EN
Graphs Comb. 3, 55-66 (1987).
00152859
1987
j
52Bxx 05A20
h-vector; centrally-symmetric simplicial polytope; Cohen-Macaulay simplicial complexes
Author's abstract: ''\textit{I. B??r??ny} and \textit{L. Lov??sz} [Acta Math. Acad. Sci. Hung. 40, 323-329 (1982; Zbl 0514.52003)] showed that a d- dimensional centrally-symmetric simplicial polytope \({\mathcal P}\) has at least \(2^ d\) facets, and conjectured a lower bound for the number \(f_ i\) of i-dimensional faces of \({\mathcal P}\) in terms of d and the number \(f_ 0=2n\) of vertices. Define integers \(h_ 0,...,h_ d\) by \(\sum^{d}_{i=0}f_{i-1}(x-1)^{d-i}= \sum^{d}_{i=0}h_ ix^{d-i}.\) A. Bj??rner conjectured (unpublished) that \(h_ i\geq \left( \begin{matrix} d\\ i\end{matrix} \right)\) (which generalizes the result of B??r??ny-Lov??sz since \(f_{d-1}=\sum h_ i)\), and more strongly that \(h_ i- h_{i-1}\geq \left( \begin{matrix} d\\ i\end{matrix} \right)- \left( \begin{matrix} d\\ i-1\end{matrix} \right),\) \(1\leq i\leq [d/2],\) which is easily seen to imply the conjecture of B??r??ny-Lov??sz. In this paper the conjectures of Bj??rner are proved.''
The proof uses Cohen-Macaulay simplicial complexes and toric varieties. The author points out that for the corresponding upper bound problem (largest possible value of \(f_ i\) for a centrally-symmetric simplicial d-polytope with \(f_ 0=2n\) vertices) not even a plausible conjecture is known.
R.Schneider
Zbl 0514.52003