Preface

The main purpose of this book is to bring together some areas of research that have developed independently over the last 30 years. The central problem we are going to discuss is that of the computation of the number of integral points in suitable families of variable polytopes. This problem is formulated in terms of the study of partition functions. The partition function TX(b), associated to a finite set of integral vectors X, counts the number of ways in which a variable vector b can be written as a linear combination of the elements in X with positive integer coefficients. Since we want this number to be finite, we assume that the vectors X generate a pointed cone C(X). Special cases were studied in ancient times, and one can look at the book of Dickson [50] for historical information on this topic. The problem goes back to Euler in the special case in which X is a list of positive integers, and in this form it was classically treated by several authors, such as Cayley, Sylvester [107] (who calls the partition function the quotity), Bell [15] and Ehrhart [53], [54]. Having in mind only the principal goal of studying the partition functions, we treat several topics but not in a systematic way, by trying to show and compare a variety of different approaches. In particular, we want to revisit a sequence of papers of Dahmen and Micchelli, which for our purposes, culminate in the proof of a slightly weaker form of Theorem 13.54, showing the quasipolynomial nature of partition functions [37] on suitable regions of space.