Stanley, Richard Generalized h-vectors, intersection cohomology of toric varieties, and related results. (English) Zbl 0652.52007 Commutative algebra and combinatorics, US-Jap. Joint Semin., Kyoto/Jap. 1985, Adv. Stud. Pure Math. 11, 187-213 (1987). [For the entire collection see Zbl 0632.00003.] The “f-vector” of a d-dimensional convex polytope is \((f_ 0,f_ 1,...,f_{d-1})\), where \(f_ i\) is the number of i-dimensional faces of the given polytope. The “h-vector” is another numerical sequence associated to a polytope, which can be defined by its f-vector. Using cohomology of projective toric varieties, the h-vectors of simplicial polytopes have been completely characterized. The author extends the notion of h-vector to “Eulerian” posets and proves that these h-vectors still satisfy the Dehn-Sommerville equations. Using intersection cohomology of projective toric varieties, he obtains results on h-vectors of rational non-simplicial polytopes. Further generalizations, several conjectures and open problems are discussed. Reviewer: F.Pauer Cited in 9 ReviewsCited in 44 Documents MSC: 52Bxx Polytopes and polyhedra 14F99 (Co)homology theory in algebraic geometry 14M25 Toric varieties, Newton polyhedra, Okounkov bodies Keywords:f-vector; h-vector; intersection cohomology of projective toric varieties; non-simplicial polytopes Citations:Zbl 0632.00003 PDFBibTeX XML