Weierstrass sets on finite graphs

TL;DR

This paper explores tropical analogues of Weierstrass semigroups on graphs, introducing rank and functional Weierstrass sets, and characterizes their properties and differences through theoretical results and constructions.

Contribution

It introduces and compares two tropical Weierstrass set concepts on graphs, providing characterizations and examples that distinguish their properties.

Findings

On simple graphs, the rank set is contained in the functional set.

Complete characterization of subsets of N as functional Weierstrass sets.

Construction of rank Weierstrass sets that are not semigroups.

Abstract

We study two possible tropical analogues of Weierstrass semigroups on graphs, called rank and functional Weierstrass sets. We prove that on simple graphs, the first is contained in the second. We completely characterize the subsets of N arising as a functional Weierstrass set of some graph. Finally, we give a sufficient condition for a subset of N to be the rank Weierstrass set of some graph, allowing us to construct examples of rank Weierstrass sets that are not semigroups.