tailieunhanh - Báo cáo toán học: " h∗-vectors, Eulerian polynomials and stable polytopes of graph"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: h∗-vectors, Eulerian polynomials and stable polytopes of graphs. | h -vectors Eulerian polynomials and stable polytopes of graphs Christos A. Athanasiadis Department of Mathematics University of Crete 71409 Heraklion Crete Greece caa@ Submitted Jul 5 2004 Accepted Sep 22 2004 Published Oct 7 2004 Mathematics Subject Classifications Primary 52B20 Secondary 05C17 05E99 13F20. Abstract Conditions are given on a lattice polytope P of dimension m or its associated affine semigroup ring which imply inequalities for the h -vector h hl . hm of P of the form h h d-i for 1 i d 2 and h d h d 2ỵ 1 h d where h 0 for d i m. Two applications to order polytopes of posets and stable polytopes of perfect graphs are included. 1 Introduction Let P be an m-dimensional convex polytope in RN having vertices with integer coordinates. It is a fundamental result due to Ehrhart 5 6 that the function i P r rP 0 ZN counting integer points in the r-fold dilate of P is a polynomial in r of degree m called the Ehrhart polynomial of P. Thus one can write tr r h h t hmtm 1 - t m 1 1 for certain integers h . Following Stanley 21 we call h hl . h m the h -vector of P and denote it by h P . It is known that i P r is the Hilbert function of a semistandard graded Cohen-Macaulay normal domain RP called the semigroup ring of P see 3 Chapter 6 and 9 Chapter X . In particular the integers h are nonnegative. Recall that a sequence a a1 . an of real numbers is said to be unimodal if a aj an holds for some 0 j n. Although h -vectors are not always unimodal various results and conjectures concerning the unimodality of h P have appeared in the literature 8 9 19 20 . For instance it would follow from 8 Conjecture and 19 Conjecture 4a that h P is unimodal if the semigroup ring RP is standard and Gorenstein. Moreover THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2 2004 R6 1 it was conjectured by Hibi 9 p. 111 that h P is unimodal whenever it is symmetric meaning that h hm-i for 0 i m. General conditions on an integer polytope P inspired by the work of V. Reiner and V.