On Sylvester sums of compound sequence semigroup complements

TL;DR

This paper generalizes the characterization of non-representable numbers in compound sequence semigroups, computes Sylvester sums, and applies these results to determine weights of higher-order Weierstrass points on algebraic curves.

Contribution

It extends Tuenter's result to compound sequence semigroups and applies this to algebraic geometry, specifically to Weierstrass point weights.

Findings

Complete characterization of non-representable numbers in compound sequence semigroups

Explicit formulas for Sylvester sums in this context

Application to weights of higher-order Weierstrass points

Abstract

In this paper, we consider the set $\textit{NR}(G)$ of natural numbers which are not in the numerical semigroup generated by a compound sequence $G$. We generalize a result of Tuenter which completely characterizes $\textit{NR}(G)$. We use this result to compute Sylvester sums, and we give a direct application to the computation of weights of higher-order Weierstrass points on some families of complex algebraic curves.