# Characterizations of numerical semigroup complements via Ap\'ery sets

## TL;DR

This paper provides a comprehensive characterization of numerical semigroup complements using Apéry sets, enabling explicit computation of Sylvester sums for free numerical semigroups and extending previous work.

## Contribution

It generalizes Tuenter's identity to characterize complements of numerical semigroups via Apéry sets and derives explicit formulas for Sylvester sums.

## Key findings

- Derived a complete characterization of semigroup complements
- Computed Sylvester sums for free numerical semigroups
- Provided explicit formulas for small powers

## Abstract

In this paper, we generalize the work of Tuenter to give an identity which completely characterizes the complement of a numerical semigroup in terms of its Ap\'ery sets. Using this result, we compute the $m$th power Sylvester and alternating Sylvester sums for free numerical semigroups. Explicit formulas are given for small $m$.