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

Helminck, Paul Alexander (2022) 'Invariants for trees of non-archimedean polynomials and skeleta of superelliptic curves.', Mathematische Zeitschrift., 301 (2). pp. 1259-1297.

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 yn=f(x). We first define a set of tropical invariants for f(x) using symmetrized Plücker coordinates and we show that these invariants determine the tree associated to f(x). This tree then 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 of degree d≤5.

Item Type:Article
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1007/s00209-021-02959-5
Publisher statement:This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
Date accepted:23 November 2021
Date deposited:17 February 2022
Date of first online publication:16 January 2022
Date first made open access:17 February 2022

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