2023/12/11 by Feihu Liu, Liu, Feihu · 1 citation
Mathematics · Computer Science · #Commutative Algebra and Its Applications #Rings, Modules, and Algebras #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2312.06096
Let ⟨ A⟩ be the numerical semigroup generated by relatively prime positive integers \a1,a2,...,an\. The quotient of ⟨ A⟩ with respect to a positive integer p is defined by (⟨ A⟩)/(p)=\x∈ ℕ | px∈ ⟨ A⟩\. The quotient (⟨ A⟩)/(p) is known to be a semigroup but is hard to study. When p is a positive divisor of a1, we reduce the computation of the Apéry set of (a1)/(p) in (⟨ A⟩)/(p) to a simple minimization problem. This allow us to obtain closed formulas of the Frobenius number of the quotient for some special numerical semigroups. These includes the cases when ⟨ A⟩ is the almost arithmetic progressions, the almost arithmetic progressions with initial gaps, etc. In particular, we partially solve an open problem proposed by A. Adeniran et al.