Classifying polynomial maps by means of polyhedral geometry

通过多面体几何对多项式映射进行分类

• 批准号: 446593912
• 负责人: Dr. Boulos El Hilany, Ph.D.
• 金额: --
• 依托单位: Institut für Analysis und Algebra
• 依托单位国家: 德国
• 项目类别: WBP Position
• 财政年份: 2020
• 资助国家: 德国
• 起止时间: 2019-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/446593912?language=en
• 关键词: Classifying; polynomial; maps; means; polyhedral

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.

在柏拉图的洞穴寓言中，人们被困在洞穴中，只能看到洞穴外真实的世界的运动影子。洞穴墙壁上的这些阴影图像构成了他们的现实，因为这是他们所知道的一切-他们不能转身看洞穴外面。类似地，对于多项式映射的拓扑，我们被限制在观察分支集--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。根据柏拉图的说法，只有极少数真正的哲学家才能成功地从洞穴中逃到现实中。现在，我们将满足于从阴影中推断。这个项目关注从复平面到其自身的多项式映射的研究。也就是说，在这些函数下的图像中的点的坐标是源空间中的点的坐标的多项式。这种映射的拓扑类型表示其原像在目标平面上的轨迹所呈现的形状。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，动力系统中的迭代多项式映射、隐式统计模型的似然函数和优化问题的解的每一个行为，如果它们各自的多项式映射在拓扑上不同的话，可能会有很大的不同，而缺乏有效的程序来区分这些情况，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歧集的准确表征。这是目标空间中原像是局部平凡纤维化的最小点集。因此，我的项目的重点是一个新的描述的分歧集。我最近设计了一个组合方法来描述的一部分集生产非典型行为以外的复杂的飞机。对于补充部分，我将采用已知的技术，如A-判别式和复曲面几何。通过合并所有这些方法，我将达到两个相互独立的描述的分歧集，设计用于不同的应用程序：首先，一个精确的表征适合于多项式结构简单。其次，本质上，四种可能的方法来转换成热带曲线的组合类型的分类问题。我将通过设计一个对应定理来实现这一点，该定理将平面曲线的拓扑与图的组合学联系起来。