Classifying polynomial maps by means of polyhedral geometry

在柏拉图的洞穴寓言中，人们被困在洞穴中，只能看到洞穴外现实世界运动的影子。对他们来说，洞穴墙壁上的影子图像构成了现实，因为这是他们所知道的一切——他们无法转身向洞穴外看。类似地，对于多项式映射的拓扑结构，我们被限制于观察分岔集——阴影——以便能够对非典型纤维——投射阴影的真实物体做出陈述。柏拉图认为，只有极少数真正的哲学家才能成功地逃离洞穴，进入现实。现在，我们将满足于从阴影中推断。这个项目涉及从复平面到自身的多项式映射的研究。也就是说，在这些函数下，图像中点的坐标是源空间中点坐标的多项式。这种映射的拓扑类型表示其原像在目标平面上的轨迹所取的形状。区分拓扑相同和拓扑不同的映射是数学中许多应用的关键。例如，动力系统中迭代多项式映射的每个行为，隐式统计模型的似然函数，以及优化问题的解决方案，如果它们各自的多项式映射在拓扑上不同，可能会有很大的不同。缺乏有效的方法来区分这些情况阻碍了多项式映射的开放拓扑分类问题的任何进展。在这个项目中，我将开发一套方法来填补这一空白。在我看来，建立这些方法的理想基础是对分岔集的准确描述。这是目标空间中原像是局部平凡颤振的最小点集。因此，我项目的重点是对分岔集的新颖描述。我最近设计了一种组合方法来描述在复平面之外产生非典型行为的集合部分。对于互补部分，我将采用已知的技术，如a判别式和环面几何。通过合并所有这些方法，我将得出两个相互独立的分岔集描述，为不同的应用而设计：首先，适合于具有简单结构的多项式的精确表征。其次，从本质上讲，有四种可能的方法将问题转化为热带曲线组合类型的分类。我将通过设计一个将平面曲线的拓扑与图的组合联系起来的对应定理来实现这一点。

In Plato's allegory of the cave, people are trapped in a cave and can only see the shadows of movements from the real world outside the cave. These shadow images on the cave's wall constitute reality to them, as it's all they know - they cannot turn around to look outside the cave. Similarly, with topology of polynomial maps, we are restricted to looking at the bifurcation set - the shadows - to be able to make statements about the atypical fibres - the true objects casting the shadows. According to Plato, only the rare true philosopher will succeed in escaping from the cave into reality. For now, we will content ourselves with extrapolating from the shadows.This project concerns the study of polynomial maps from the complex plane to itself. That is, the coordinates of points in the image under those functions are polynomials in the coordinates of points in the source space. The topological type of such a map denotes the shape taken by its preimages' locus over the target plane.Distinguishing between topologically identical and topologically different maps is key for numerous applications in mathematics. For instance, each of the behaviours of iterated polynomial maps in dynamical systems, likelihood functions of an implicit statistical model, and solutions to optimization problems may vary considerably if their respective polynomial maps differ in topology.The lack of effective procedures to differentiate between those cases hinders any progress in the open topological classification problem of polynomial maps. In this project, I will develop a set of methods to fill this gap.In my opinion, an ideal base on which to build these approaches is an accurate characterization of the bifurcation set. That is the smallest set of points in the target space at which the preimage is a locally trivial fibration. Hence, the focal point of my project is a novel description of the bifurcation set. I have recently designed a combinatorial method to describe the part of the set producing atypical behaviour outside the complex plane. For the complementary part, I will adapt known techniques such as A-discriminants and toric geometry. By merging all these approaches, I will arrive at two mutually independent descriptions of the bifurcation set, designed for distinct applications:Firstly, a precise characterization suited for polynomials with simple structures. Secondly and essentially, four possible approaches to transform the problem into a classification of combinatorial types of tropical curves. This I will achieve by designing a correspondence theorem linking the topology of planar curves with the combinatorics of graphs.