Generalising Kapranov's Theorem For Tropical Geometry Over Hyperfields

Abstract

Kapranov's theorem is a foundational result in tropical geometry. It states that the set of tropicalisations of points on a hypersurface coincides precisely with the tropical variety of the tropicalisation of the defining polynomial. The aim of this paper is to generalise Kapranov's theorem, replacing the role of a valuation map, from a field to the real numbers union negative infinity, with a more general class of hyperfield homomorphisms, whose target is the tropical hyperfield and satisfy a relative algebraic closure condition. To provide an example of such a hyperfield homomorphism, the map from the complex tropical hyperfield to the tropical hyperfield is investigated. There is a brief outline of sufficient conditions for a hyperfield homomorphism to satisfy the relative algebraic closure condition.