Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: The h-vector of a Gorenstein toric ring of a compressed polytope. | The -vector of a Gorenstein toric ring of a compressed polytope Hidefumi Ohsugi Department of Mathematics Faculty of Science Rikkyo University Toshima Tokyo 171-8501 Japan ohsugi@ Takayuki Hibi Department of Pure and Applied Mathematics Graduate School of Information Science and Technology Osaka University Toyonaka Osaka 560-0043 Japan hibi@ Submitted May 8 2005 Accepted August 9 2005 Published October 1 2005 Mathematics Subject Classifications 52B20 13H10 Dedicated to Richard P. Stanley on the occasion of his 60th birthday Abstract A compressed polytope is an integral convex polytope all of whose pulling triangulations are unimodular. A q 1 -simplex E each of whose vertices is a vertex of a convex polytope P is said to be a special simplex in P if each facet of P contains exactly q 1 of the vertices of E. It will be proved that there is a special simplex in a compressed polytope P if and only if its toric ring K P is Gorenstein. In consequence it follows that the -vector of a Gorenstein toric ring K P is unimodal if P is compressed. A compressed polytope 10 p. 337 is an integral convex polytope all of whose pulling triangulations are unimodular. Recall that an integral convex polytope is an convex polytope each of whose vertices has integer coordinates. A typical example of compressed polytopes is the Birkhoff polytopes 10 Example b . Later in 6 a large class of compressed polytopes including the Birkhoff polytopes is presented. Recently Seth Sullivant 12 proved a surprising result that the class given in 6 does essentially contain all compressed polytopes. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2 2005 N4 1 Let P c Rn be an integral convex polytope. Let K be a field and K x x-1 t K x1 x-1 . xn x-1 t the Laurent polynomial ring in n 1 variables over K. The toric ring of P is the subalgebra K P of K x x-1 t which is generated by those monomials xat x 1 xf t such that a a1 . an belongs to P Pl Zn. We will regard K P as
