Computational Tropical Geometry and its Applications

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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.