Computational Tropical Geometry and its Applications

在高中，学生们学习到方程x^2-x-2=0有两个解：x=-1和x=2。但是如果有更多的方程和变量，比如x^2+11*y^2+x+3 y +1 = 0和x^2+13*y^2+2x+y+5 = 0，情况又会怎样呢？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沿着装配线的运动。求解这类方程组是一项重要但计算困难的任务，同伦延拓是求解这类方程组的一种常用方法，它涉及到选择一个具有已知解的简单方程组，并将其变形为目标方程组。如果仔细操作，启动系统的每个解都可以追溯到目标系统的解。对于上述方程组，其起始方程组与目标方程组的可能解为x^2-x-2 = 0和y^2-y-2 = 0，但选择起始方程组需要准确估计目标解的个数.这对于在应用中出现的系统尤其具有挑战性，其中变量具有意义并且方程具有结构。由于不同应用的意义和结构各不相同，很难找到一种通用的方法。这是我们将使用热带几何的主要项目。在热带几何中，每个多项式都被分配了一个分段线性对象，并且从这些所谓的热带品种如何彼此相交，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，如遗传学，经济学或机器学习。这种多样性是热带几何学的主要优势之一，我们将加以利用。它是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。

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.