Star-shaped Complexes and Ehrhart Polynomials

We study Ehrhart polynomials of star-shaped triangulations of balls by means of Cohen-Macaulay rings and canonical modules. A polyhedral complex F in RN is a finite set of convex polytopes in RN such that (1.1) if & £ T and & is a face of &, then & e I\ and (1.2) if &, & £ T, then 9° n 0, write nX for {na;a £ X] and define i(X, n) tobe #(nXC\1N), the cardinality of nXr\ZN. In other words, i(X, n) is equal to the number of rational points (ax, a2, ... , aN) £ X with each «a, £ Z. It is known that (3.1) i(X, n) is a polynomial in « of degree d, called the Ehrhart polynomial of X, (3.2) i(X,0) = l,and (3.3) (-l)di(X,-n) = #[n(X-dX)f\ZN] for every l<«eZ. Define the sequence So, ô\, ô2, ... of integers by the formula (l-X) d+\ = YwL n=\ J 1=0 Then (4.1) ¿o=l and Sx = #(XnZN)-(d+l), (4.2) d, and (4.3) 6d = #[(X dX) n ZN]. We say that ô(X) (ô0, ¿i, ... , ôj) is the â-vector of X. We refer the reader to, e.g., [6, Chapter IX], for geometric proofs of the above fundamental results Received by the editors June 18, 1993; this paper was presented in the meeting "Combinatorial Convexity and Algebraic Geometry" held at Mathematisches Forschungsinstitut Oberwolfach, March 28-April 3, 1993. 1991 Mathematics Subject Classification. Primary 13D40. © 1995 American Mathematical Society 0002-9939/95 $1.00+ $.25 per page