2023/06/20 by Tristram Bogart, Bogart, Tristram, Christopher O’Neill +3
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2306.11564
openalex publication_date 2023/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We examine two natural operations to create numerical semigroups. We say that a numerical semigroup S is k-normalescent if it is the projection of the set of integer points in a k-dimensional polyhedral cone, and we say that S is a k-quotient if it is the quotient of a numerical semigroup with k generators. We prove that all k-quotients are k-normalescent, and although the converse is false in general, we prove that the projection of the set of integer points in a cone with k extreme rays (possibly lying in a dimension smaller than k) is a k-quotient. The discrete geometric perspective of studying cones is useful for studying k-quotients: in particular, we use it to prove that the sum of a k1-quotient and a k2-quotient is a (k1+k2)-quotient. In addition, we prove several results about when a numerical semigroup is not k-normalescent.