tailieunhanh - Báo cáo toán học: "Counting d-polytopes with d + 3 vertices"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Counting d-polytopes with d + 3 vertices. | Counting d-polytopes with d 3 vertices Eric Fusy Algorithm Project INRIA Rocquencourt France Submitted Nov 23 2005 Accepted Mar 5 2006 Published Mar 14 2006 Mathematics Subject Classifications 52B11 52B35 05A15 05A16 Abstract We completely solve the problem of enumerating combinatorially inequivalent d-dimensional polytopes with d 3 vertices. A first solution of this problem by Lloyd was published in 1970. But the obtained counting formula was not correct as pointed out in the new edition of Grunbaum s book. We both correct the mistake of Lloyd and propose a more detailed and self-contained solution relying on similar preliminaries but using then a different enumeration method involving automata. In addition we introduce and solve the problem of counting oriented and achiral . stable under reflection d-polytopes with d 3 vertices. The complexity of computing tables of coefficients of a given size is then analyzed. Finally we derive precise asymptotic formulas for the numbers of d-polytopes oriented d-polytopes and achiral d-polytopes with d 3 vertices. This refines a first asymptotic estimate given by Perles. 1 Introduction A polytope P is the convex hull of a finite set of points of a vector space Rd. If P is not contained in any hyperplane of Rd then P is said d-dimensional or is called a d-polytope. A vertex resp. a facet of P is defined as the intersection of P with an hyperplane H of Rd such that P n H has dimension 0 resp. has dimension d 1 and one of the two open sides of H does not meet P. A vertex v is incident to a facet f if v 2 f. This article addresses the problem of counting combinatorially different d-polytopes with d 3 vertices meaning that two polytopes are identified if their incidences vertices-facets are isomorphic . the incidences are the same up to relabeling of the vertices . Under this equivalence relation polytopes are refered to as combinatorial polytopes. Whereas general d-polytopes are involved objects .