The Fundamental Theorem of Tropical Geometry over Hyperfields

I am exploring the applications of the Fundamental Theorem of Tropical Geometry, and trying to understand whether this relationship between the Tropicalisation of polynomials and the solutions of these polynomials holds under different conditions. The setting I am applying this theorem to is the hyperfield setting. Hyperfields are a generalised idea of our normal algebraic object fields, where the addition operation is allowed to be multivalued.

To begin with I am exploiting the Krasner construction of Hyperfields, using a ring under the quotient of a subset of the multiplicative subgroup. This allows us to generate hyperfields where the operation is multivalued, built in by the construction. Then by using the Matroid structure from Baker and Bowler we are looking to apply the theory from tropical geometry to that of Hyperfields, in terms of the circuits of the matroids.