2017/12/07 by Alexandru Pascadi, Pascadi, Alexandru
Computer Science · Mathematics · #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Graph theory and applications #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1712.02522
openalex publication_date 2017/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper proposes a new, visual method to study numerical semigroups and the Frobenius problem. The method is based on building a so-called reduction graph, whose nodes usually correspond to monogenic semigroups, and whose edges can have multiple inputs and outputs. If such a construction is possible, then determining whether the studied semigroup is symmetric, or finding explicit forms of its Apéry set and Hilbert series, is reduced to straightforward computations assisted by a MAPLE program we made available on arXiv. This approach applies to many of the cases considered in literature, including semigroups generated by arithmetic and geometric sequences, compound sequences, progressions of the form an, an + a, …, an + an-1, triangular and tetrahedral numbers, certain Fibonacci triplets, etc. After explaining the general approach in more detail, the paper studies the types of edges that can be used as building blocks of a reduction graph, as well as a series of operations that serve to modify or combine valid reduction graphs. In the end of the paper, we use these techniques to solve the Frobenius problem for 7 new classes of numerical semigroups.