Let \(f_1(n), \ldots , f_k(n)\) be polynomial functions of n. For fixed \(n\in \mathbb {N}\), let \(S_n\subseteq \mathbb {N}\) be the numerical semigroup generated by \(f_1(n),\ldots ,f_k(n)\). 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 \(S_n\subseteq \mathbb {N}^m\) 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.

令\（f_1（n），\ ldots，f_k（n）\）是n的多项式函数。对于固定的\（n \ in \ mathbb {N} \），令\（S_n \ subseteq \ mathbb {N} \）为\（f_1（n），\ ldots，f_k（n）\）生成的数值半群。。由于ñ变化，我们展示了许多不变量\（S_N \）是最终准多项式ñ，最引人注目的是贝蒂数，而且类型，属，和大小\（\三角洲\） -set 。我们使用的工具是参数化Presburger算术逻辑系统中的可表达性。推广到半群的高维族，我们还检验了仿射半群\（S_n \ subseteq \ mathbb {N} ^ m \）由其坐标是n的多项式函数的向量生成，并且我们证明在这种情况下Betti数最终还是n的拟多项式函数。