Periodic behavior in families of numerical and affine semigroups via parametric Presburger arithmetic

Let f(1)(n), ... , f(k) (n) be polynomial functions of n. For fixed n is an element of N, let S-n subset of N be the numerical semigroup generated by f(1)(n), ... , f(k) (n). As n varies, we show that many invariants of S-n are eventually quasi-polynomial in n, most notably the Betti numbers, but also the type, the genus, and the size of the Delta-set. The tool we use is expressibility in the logical system of parametric Presburger arithmetic. Generalizing to higher dimensional families of semigroups, we also examine affine semigroups S-n subset of N-m generated by vectors whose coordinates are polynomial functions of n, and we prove that in this case the Betti numbers are also eventually quasi-polynomial functions of n