论文标题
热带平面中的点线几何形状
Point-Line Geometry in the Tropical Plane
论文作者
论文摘要
我们研究了Bruijn和Erd \ h OS的经典结果，这些结果是在飞机上由$ N $ - 点配置确定的线的界限，以及最近经过证明的热带Sylvester-Gallai定理，提出了上述结果的热带版本。在这项工作中，我们引入了稳定的热带线，这有助于回答与热带平面中发病率几何形状有关的问题。热带平面中的投射二元性有助于将问题转换为稳定的线路，以将其深入研究的稳定交叉点。借助牛顿细分和线条排列之间的二元性，我们能够将稳定的交叉点与细胞形状的分类分类，这最终有助于我们提出界限。在此过程中，我们还遇到了线性牛顿细分线的各种独特属性，这些属性是对热带线条的双重布置。
We study the classical result by Bruijn and Erd\H os regarding the bound on the number of lines determined by a $n$-point configuration in the plane, and in the light of the recently proven Tropical Sylvester-Gallai theorem, come up with a tropical version of the above-mentioned result. In this work, we introduce stable tropical lines, which help in answering questions pertaining to incidence geometry in the tropical plane. Projective duality in the tropical plane helps in translating the question for stable lines to stable intersections that have been previously studied in depth. Invoking duality between Newton subdivisions and line arrangements, we are able to classify stable intersections with shapes of cells in subdivisions, and this ultimately helps us in coming up with a bound. In this process, we also encounter various unique properties of linear Newton subdivisions which are dual to tropical line arrangements.
