Submission date: 20 November 2019

Abstract:

Let f_1(n), ..., f_k(n) be polynomial functions of n. For fixed n in ℕ, let S_n ⊆ 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, such as the Frobenius number, the type, the genus, and the size of the Δ-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 ⊆ ℕ^m generated be vectors whose coordinates are polynomial functions of n, and we prove similar results; for example, the Betti numbers are eventually quasi-polynomial functions of n.

Mathematics Subject Classification: 20M14, 05A15, 52B20

Keywords and phrases:

Full text arXiv 1911.09136: pdf, ps.