Hyperfields for Tropical Geometry I. Hyperfields and dequantization

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.

47 pages, 5 figures, the previous version has been radically changed in order to add new references and correct terminology

References in corpus (2)

• Tropical Geometry and its applications
• Correspondence principle for idempotent calculus and some computer applications

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• Symmetrization of monoïds as hypergroups
• Commutative $\nu$-algebra and supertropical algebraic geometry
• The tropical analogue of the Helton-Nie conjecture is true
• A Newton Polygon Rule for Formally-Real Valued Fields and Multiplicities over the Signed Tropical Hyperfield
• Symmetric Polynomials in Tropical Algebra Semirings
• Tropical compound matrix identities
• Matroids over Skew Tracts
• Promiscuously Quadratic Rings
• Lie pairs
• Signed tropical convexity
• Oriented Matroids from Triangulations of Products of Simplices