Geometry of tropical extensions of hyperfields

We study the geometry of tropical extensions of hyperfields, including the ordinary, signed and complex tropical hyperfields. We introduce the framework of ‘enriched valuations’ as hyperfield homomorphisms to tropical extensions, and show that a notable family of them are relatively algebraically closed. Our main results are hyperfield analogues of Kapranov’s theorem and the Fundamental theorem of tropical geometry. Utilising these theorems, we introduce fine tropical varieties and prove a structure theorem for them in terms of their initial ideals.