Invariants for trees of non-archimedean polynomials and skeleta of superelliptic curves

TL;DR

This paper extends the $j$-invariant criterion from elliptic curves to superelliptic curves, introducing tropical invariants that determine the reduction type via associated trees, with explicit descriptions for low-degree cases.

Contribution

It introduces tropical invariants for superelliptic curves that generalize elliptic curve invariants and relate these to reduction types through combinatorial trees.

Findings

Tropical invariants determine the associated tree of the polynomial.

The tree completely determines the reduction type of the superelliptic curve.

Explicit half-space conditions are provided for polynomials of degree up to 5.

Abstract

In this paper we generalize the $j$-invariant criterion for the semistable reduction type of an elliptic curve to superelliptic curves $X$ given by $y^{n}=f(x)$. We first define a set of tropical invariants for $f(x)$ using symmetrized Pl\"{u}cker coordinates and we show that these invariants determine the tree associated to $f(x)$. We then prove that this tree completely determines the reduction type of $X$ for $n$ that are not divisible by the residue characteristic. The conditions on the tropical invariants that distinguish between the different types are given by half-spaces as in the elliptic curve case. These half-spaces arise naturally as the moduli spaces of certain Newton polygon configurations. We give a procedure to write down their equations and we illustrate this by giving the half-spaces for polynomials $f(x)$ of degree $d\leq{5}$.