Rational generating functions and lattice point sets.

Rational Generating Functions and Lattice Point Sets by Kevin M. Woods Chair: Alexander Barvinok We prove that, for any fixed d, there is a polynomial time algorithm for computing the generating function of any projection of the set of integer points in a d-dimensional polytope. This implies that many interesting sets of integer points can be encoded as short rational generating functions, such as the Frobenius semigroup of all nonnega- tive integer combinations of given positive integers, affine semigroups, neighbors and the neighborhood complex (also known as the Scarf complex or complex of maximal lattice-free bodies), Hilbert bases, and sets from algebraic integer programming. We also show how to use the generating functions to solve computational problems (such as finding the cardinality of the set or finding its maximum element) in polynomial time. We may also use this theorem to compute, as a short rational function, the Hilbert series of rings generated by monomials. We examine the connection between generating functions and the neighborhood complex, and we consider possibilities for improving the algorithm for the main theorem. Finally, we examine the relationship between rational generating functions and the complexity of Presburger arithmetic.