On parametrized families of numerical semigroups. Issue 11 (1st November 2020)

• Record Type: Journal Article
• Authors: Kerstetter, Franklin; O'Neill, Christopher
• Is Part Of: Communications in algebra. Volume 48:Issue 11(2020)
• Page Start: 4698
• Page End: 4717
• Publication Date: 2020-11-01
• Subjects: Betti number -- Frobenius number -- numerical semigroup -- quasipolynomial; Primary: 20M14 -- 05E40; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2020.1769122
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: A numerical semigroup is an additive subsemigroup of the non-negative integers. In this article, we consider parametrized families of numerical semigroups of the form P n = 〈 f 1 ( n ), …, f k ( n ) 〉 for polynomial functions fi . We conjecture that for large n, the Betti numbers, Frobenius number, genus, and type of Pn 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 = 〈 n, n + r 2, …, n + r k 〉 . Our main result is to prove our conjecture in the case where each fi 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.