Hyperfields for Tropical Geometry I. Hyperfields and dequantization

TL;DR

This paper introduces new hyperfields with multivalued addition, connecting classical number sets to tropical geometry through dequantization, and lays the groundwork for their application in tropical geometry.

Contribution

It defines and studies new hyperfields based on classical number sets with multivalued addition, establishing their relation to classical fields via dequantization.

Findings

Hyperfields with multivalued addition are constructed from classical number sets.

These hyperfields are related to classical fields through dequantization processes.

The complex tropical hyperfield is a dequantization of the complex numbers.

Abstract

New hyperfields, that is fields in which addition is multivalued, are introduced and studied. In a separate paper these hyperfields are shown to provide a base for the tropical geometry. The main hyperfields considered here are classical number sets, such as the set of complex numbers, the set of real numbers, and the set of real non-negative numbers, with the usual multiplications, but new, multivalued additions. The new hyperfields are related with the classical fields and each other by dequantisations. For example, the new complex tropical field is a dequantization of the field of complex numbers.