The distribution of Weierstrass points on a tropical curve

TL;DR

This paper investigates the distribution of Weierstrass points on metric graphs, showing they become uniformly distributed according to a canonical measure as the degree of divisors increases.

Contribution

It establishes a distribution result for Weierstrass points on metric graphs, connecting combinatorial properties with potential theory and extending known results to tropical curves.

Findings

Number of Weierstrass points is g(n-g+1) for a generic divisor of degree n on a genus g metric graph.

Weierstrass points become distributed according to the Zhang canonical measure as degree grows.

Distribution relates to effective resistances on the metric graph.

Abstract

We show that on a metric graph of genus $g$, a divisor of degree $n$ generically has $g(n-g+1)$ Weierstrass points. For a sequence of generic divisors on a metric graph whose degrees grow to infinity, we show that the associated Weierstrass points become distributed according to the Zhang canonical measure. In other words, the limiting distribution is determined by effective resistances on the metric graph. This distribution result has an analogue for complex algebraic curves, due to Neeman, and for curves over non-Archimedean fields, due to Amini.