Solving Lattice Point Problems Using Rational Generating Functions

Here are two examples of questions we'd like to answer: given coprime integers a_1,a_2,...,a_d, what is the largest integer which cannot be written as a nonnegative integer combination of these a_i (this is the classic problem of Frobenius)? How many positive integers cannot be written in this way? Not only do we want to answer these questions, but we want to answer them _quickly_. In this talk, we will present several rational generating function tools which can be used to solve these and many other problems quickly (to be precise, in polynomial time for fixed d). For example, we can quickly compute the Hilbert basis of a rational cone (as a rational generating function) or the Hilbert series of a polynomial ring generated by monomials.