# Tropical geometry over the tropical hyperfield

## TL;DR

This paper develops a new framework for tropical scheme theory by integrating tropical hyperfields with ordered blueprints, enabling a scheme-theoretic approach to tropicalization and linking it to Berkovich analytification.

## Contribution

It introduces a novel scheme-theoretic tropicalization method using tropical hyperfields and ordered blueprints, connecting classical varieties with tropical geometry.

## Key findings

- Characterizes Berkovich analytification via scheme-theoretic tropicalizations.
- Shows Giansiracusa bend relations derive from scheme-theoretic tropicalization.
- Provides a topological description of tropicalization and analytification.

## Abstract

In this text, we merge ideas around the tropical hyperfield with the theory of ordered blueprints to give a new formulation of tropical scheme theory. The key insight is that a nonarchimedean absolute value can be considered as a morphism into the tropical hyperfield. In turn, ordered blueprints make it possible to consider the base change of a classical variety to the tropical hyperfield. We call this base change the scheme theoretic tropicalization of the classical variety. Our first main result describes the Berkovich analytification and the tropicalization of a classical variety as sets of rational points of scheme theoretic tropicalizations, including a characterization of the respective topologies. Our second main result shows that the Giansiracusa bend relations can be derived by a natural construction from the scheme theoretic tropicalization.