Tropical superelliptic curves

TL;DR

This paper introduces an algorithm to compute the Berkovich skeleton of superelliptic curves over valued fields, characterizes realizability of associated metric graphs, and explores their moduli space location.

Contribution

It provides a new algorithm for Berkovich skeleton computation and establishes realizability conditions for superelliptic metric graphs.

Findings

Algorithm for Berkovich skeleton computation

Realizability of superelliptic graphs when n is prime

Analysis of superelliptic graphs in the moduli space

Abstract

We present an algorithm for computing the Berkovich skeleton of a superelliptic curve $y^n=f(x)$ over a valued field. After defining superelliptic weighted metric graphs, we show that each one is realizable by an algebraic superelliptic curve when $n$ is prime. Lastly, we study the locus of superelliptic weighted metric graphs inside the moduli space of tropical curves of genus $g$.