Generalising Kapranov's theorem for tropical geometry over hyperfields
James Maxwell · Journal of Algebra · 2023
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, ν:K→R∪{−∞}, with a more general class of hyperfield homomorphisms, H→T, which satisfy a relative algebraic closure condition. The map η:TC→T, η(x):=log(|x|), where TC is the tropical complex hyperfield, provides an example of such a hyperfield homomorphism.