Short Rational Generating Functions for Lattice Point Problems

be the set of all non-negative integer combinations of a1, . . . , ad, or, in other words, the semigroup S ⊂ Z+ of non-negative integers generated by a1, . . . , ad. What does S look like? In particular, what is the largest integer not in S? (It is well known and easy to see that all sufficiently large integers are in S.) How many positive integers are not in S? How many positive integers within a particular interval or a particular arithmetic progression are not in S? One of the results of our paper is that for any fixed d “many” of these and similar questions have “easy” solutions. For some of these questions, notably, how to find the largest integer not in S, an efficient solution is already known [K92]. For others, for example, how to find the number of positive integers not in S, an efficient solution was not previously known. With a subset S ⊂ Z+ we associate the generating function f(S;x) = ∑