The Fundamental Theorem of Tropical Geometry over Hyperfields

超场热带几何基本定理

• 批准号: 2236572
• 金额: --
• 依托单位: Swansea University
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2236572
• 关键词: Fundamental; Theorem; Tropical; Geometry; over

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.

我正在探索热带几何基本定理的应用，并试图了解多项式的热带化与这些多项式的解之间的这种关系在不同条件下是否成立。我应用这个定理的设置是超场设置。超域是普通代数对象域的广义概念，其中加法运算允许是多值的。首先，我利用超域的克拉斯纳构造，使用乘法子群的子集的商下的环。这使我们能够生成超域，其中操作是多值的，由构造内置。然后，通过使用 Baker 和 Bowler 的拟阵结构，我们希望根据拟阵电路，将热带几何理论应用到超场理论。