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条件下是否成立。我要应用这个定理的是超场设置。超域是普通代数对象域的广义概念，其中的加法运算可以是多值的。首先，我将利用超域的Krasner构造，使用乘子群子集商下的环。这允许我们生成超域，其中操作是多值的，由构造内置。然后，通过使用Baker和Bowler的矩阵结构，我们希望将热带几何理论应用到超场理论中，就矩阵的电路而言。

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.