Enumeration of geometric Weierstrass points of metric graphs

TL;DR

This paper extends the concept of Weierstrass points to metric graphs, proving that generic graphs of genus g have g^3 - g such points, paralleling classical algebraic curve results.

Contribution

It introduces geometric Weierstrass points for metric graphs and establishes their count for generic graphs, providing a new proof of their existence for genus ≥ 2.

Findings

Generic metric graphs of genus g have g^3 - g geometric Weierstrass points.

The methods offer a new proof of Weierstrass points' existence on metric graphs.

The results parallel classical algebraic geometry theorems.

Abstract

A classical result states that on a smooth algebraic curve of genus $g$ the number of Weierstrass points, counted with multiplicity, is $g^3-g$. In this paper, we introduce the notion of geometric Weierstrass points of metric graphs and show that a generic metric graph of genus $g$ has $g^3-g$ geometric Weierstrass points counted with multiplicity. Our methods also provide a new proof of the existence of Weierstrass points on metric graphs of genus bigger than or equal to $2$.