On parametrized families of numerical semigroups

TL;DR

This paper investigates parametrized families of numerical semigroups generated by polynomial functions, proving that key invariants like Betti numbers and Frobenius number are quasipolynomials when these functions are linear.

Contribution

It proves the conjecture for linear polynomial generators and introduces weighted factorization length, extending known results in factorization theory.

Findings

Betti numbers and Frobenius number are quasipolynomials for large n.

Proved the conjecture for families with linear polynomial generators.

Developed the concept of weighted factorization length.

Abstract

A numerical semigroup is an additive subsemigroup of the non-negative integers. In this paper, we consider parametrized families of numerical semigroups of the form $P_n = \langle f_1(n), \ldots, f_k(n) \rangle$ for polynomial functions $f_i$. We conjecture that for large $n$, the Betti numbers, Frobenius number, genus, and type of $P_n$ each coincide with a quasipolynomial. This conjecture has already been proven in general for Frobenius numbers, and for the remaining quantities in the special case when $P_n = \langle n, n + r_2, \ldots, n + r_k \rangle$. Our main result is to prove our conjecture in the case where each $f_i$ is linear. In the process, we develop the notion of weighted factorization length, and generalize several known results for standard factorization lengths and delta sets to this weighted setting.