The Fundamental Theorem of Tropical Geometry over Hyperfields

我正在探索热带几何基本定理的应用，并试图理解多项式的热带化和这些多项式的解之间的这种关系在不同的条件下是否成立。我应用这个定理的背景是超场背景。超场是我们的正常代数对象场的一般概念，其中的加法运算被允许是多值的。首先，我利用超域的Krasner构造，使用乘法子群的一个子集的商下的环。这允许我们生成超域，其中的操作是多值的，由构造内置。然后，通过使用贝克和鲍勒的拟阵结构，我们希望将热带几何的理论应用到超场的理论中，就拟阵的回路而言。

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.