The Realization Problem for Delta Sets of Numerical Semigroups

TL;DR

This paper investigates which finite sets can be realized as delta sets of numerical semigroups, providing constructions for certain two-element sets and characterizing their occurrence in embedding dimension three.

Contribution

It characterizes the realizability of two-element delta sets in numerical semigroups and constructs explicit examples, especially focusing on embedding dimension three.

Findings

Two-element delta sets $oxed{ ext{d,td}}$ are realizable for certain conditions.

For $t eq 2$, $oxed{ ext{d,td}}$ cannot occur in embedding dimension three.

Necessary condition $oxed{ ext{min} \Delta(S) = ext{gcd} \Delta(S)}$ is established.

Abstract

The delta set of a numerical semigroup $S$, denoted $\Delta(S)$, is a factorization invariant that measures the complexity of the sets of lengths of elements in $S$. We study the following problem: Which finite sets occur as the delta set of a numerical semigroup $S$? It is known that $\min \Delta(S) = \gcd \Delta(S)$ is a necessary condition. For any two-element set $\{d,td\}$ we produce a semigroup $S$ with this delta set. We then show that for $t\ge 2$, the set $\{d,td\}$ occurs as the delta set of some numerical semigroup of embedding dimension three if and only if $t=2$.