Computational Tropical Geometry and its Applications

计算热带几何及其应用

• 批准号: MR/Y003888/1
• 负责人: Yue Ren
• 金额: $ 63.15万
• 依托单位: Durham University
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2024
• 资助国家: 英国
• 起止时间: 2024 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=MR%2FY003888%2F1
• 关键词: Computational; Tropical; Geometry; its; Applications

In high school, students learn that the equation x^2-x-2=0 has two solutions: x=-1 and x=2. But what if there are more equations and variables like x^2+11*y^2+x+3y+1 = 0 and x^2+13*y^2+2x+y+5 = 0?Such polynomial systems appear in many applications, they describe the dynamics of reactions in a chemical production process or the movement of robot arms along an assembly line. Solving these systems is an important but computationally difficult task.A go-to method for solving such systems is homotopy continuation, which involves picking an easy start system with known solutions and deforming it to the target system. If done carefully, every solution of the start system can be traced to a solution of the target system. For the system above, a potential start system to the target system above is x^2-x-2 = 0 and y^2-y-2 = 0.However, picking the start system requires an accurate estimate on the number of target solutions. This is especially challenging for systems arising in applications, where the variables carry meaning and the equations have structure. Because meaning and structure vary from application to application, it is difficult to find a method that works in general.This is the main project for which we will be using using tropical geometry. In tropical geometry, each polynomial is assigned a piecewise linear object, and from how these so-called tropical varieties intersect each other, we can estimate the number of target solutions. Tropical varieties also arise naturally in many other areas, like phylogenetics, economics or machine learning. This diversity is one of the key strengths of tropical geometry which we will be exploiting. It serves as a bridge between vastly different areas, allowing for an exchange of ideas and solutions. Problems that appear impossible in one area may very well be solvable in another.

在高中，学生们学到方程x^2-x-2=0有两个解：x=-1和x=2。但如果有更多的方程和变量，如x^2+11*y^2+x+3y+1=0和x^2+13*y^2+2x+y+5=0呢？这样的多项式系统出现在许多应用中，它们描述了化学生产过程中的反应动力学或机器人手臂在装配线上的运动。求解这类系统是一项重要但计算困难的任务，求解这类系统的一种常用方法是同伦延拓法，即选取一个有已知解的容易启动的系统并将其变形为目标系统。如果仔细操作，启动系统的每一个解决方案都可以追溯到目标系统的一个解决方案。对于上述系统，目标系统的潜在启动系统为x^2-x-2=0和y^2-y-2=0。但是，选择启动系统需要对目标解的数量进行准确估计。这对于在应用中出现的系统来说尤其具有挑战性，在这些应用中，变量具有意义，而方程具有结构。由于意义和结构因应用而异，很难找到一种普遍适用的方法。这是我们将使用热带几何进行的主要项目。在热带几何中，每个多项式都被分配了一个分段线性对象，从这些所谓的热带品种如何相互交叉，我们可以估计目标解的数量。热带品种也自然出现在许多其他领域，如系统发育学、经济学或机器学习。这种多样性是热带几何学的关键优势之一，我们将加以利用。它是巨大不同领域之间的桥梁，允许交流想法和解决方案。在一个领域看起来不可能的问题，在另一个领域很可能是可以解决的。