A Graphical Approach to Finding the Frobenius Number, Genus and Hilbert Series of a Numerical Semigroup

TL;DR

This paper introduces a visual reduction graph method to analyze numerical semigroups, simplifying the computation of Frobenius numbers, genus, and Hilbert series, and applies it to various classes including Fibonacci and arithmetic sequences.

Contribution

The paper presents a novel graphical approach for studying numerical semigroups, enabling straightforward computation of key invariants and solving the Frobenius problem for new classes.

Findings

Developed a reduction graph method for numerical semigroups.

Implemented a MAPLE program to assist computations.

Solved the Frobenius problem for 7 new classes of semigroups.

Abstract

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 Ap\'ery 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 $a^n, a^n + a, \ldots, a^n + a^{n-1}$, 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.