Tropical varieties, maps and gossip
Tropical varieties, maps and gossip
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热带品种、地图和八卦
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发表时间：
2008
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通讯作者：
Bj Frenk
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作者：
Bj Frenk
热带几何是研究热带化图的一个相对较新的数学领域：将某种类型的多面体复合体，称为热带变种，赋给嵌入的代数变种的地图。在某种意义上，它将代数几何语句转换为组合几何语句。热带几何学的一个有趣特征是，在热带变种之间不存在好的态射或地图的概念，这使得热带化地图发挥了作用。本文的主要部分研究了不同类别的热带变种之间的映射：热带线性空间和嵌入的非旋回变种的热带化。第一章是对热带几何的简要介绍。收集并证明了主要定理。这些结果都不是新的。第二章讨论嵌入的单调变种的热带化。我们给出了这类簇存在一个(不一定是内射的)参数化的充分条件，其朴素的热带化是满射到伴随的热带簇上的。第三章概述了与热带线性空间相关的代数。场和向量空间是线性代数的中心对象，半域上的半域和半域上的模也是热带线性代数和热带线性空间研究的中心对象。本章中的大多数结果都是以某种形式已知的，但分散在现有的文献中。本章的主要目的是收集这些结果，并确定足以给半域上的线性代数以熟悉的感觉的代数条件。例如，在什么条件下，线性多项式在加法和标量乘法下是闭合的？第四章是论文的主体部分。所使用的技术是热带线性代数和拟阵理论的组合。中心物品是由安德烈亚斯·德雷兹和沃尔特·温泽尔介绍的有价值的拟阵。在其他内容中，本章包含了热带线性空间上的函数的分类，该空间的循环是热带线性子空间，推广了Henry CRapo关于拟阵的初等扩张的一个旧结果。它使用Mikhalkin的热带修正概念来定义对象都是热带线性空间的范畴中的态射。最后，我们确定了从仿射2-空间到其自身的态射的开子么半群的结构为多面体复形。最后，第五章也就是最后一章只是间接地与地图有关。研究了正交群的热带化所包含的某种么半群：由距离矩阵在热带矩阵乘法下生成的么半群(即用最小代替加法，用加法代替乘法)。这个么半群将构成众所周知的八卦问题的么半群概括为这样一种环境，在这种环境中，信息的传输只有一定程度的准确性。我们确定了这个所谓的4阶矩阵的八卦么半群，并证明了它一般是一个维度等于正交群的多面体么半群。
Tropical geometry is a relatively new field of mathematics that studies the tropicalization map: a map that assigns a certain type of polyhedral complex, called a tropical variety, to an embedded algebraic variety. In a sense, it translates algebraic geometric statements into combinatorial ones. An interesting feature of tropical geometry is that there does not exist a good notion of morphism, or map, between tropical varieties that makes the tropicalization map functorial. The main part of this thesis studies maps between different classes of tropical varieties: tropical linear spaces and tropicalizations of embedded unirational varieties. The first chapter is a concise introduction to tropical geometry. It collects and proves the main theorems. None of these results are new. The second chapter deals with tropicalizations of embedded unirational varieties. We give sufficient conditions on such varieties for there to exist a (not necessarily injective) parametrization whose naive tropicalization is surjective onto the associated tropical variety. The third chapter gives an overview of the algebra related to tropical linear spaces. Where fields and vector spaces are the central objects in linear algebra, so are semifields and modules over semifields central to tropical linear algebra and the study of tropical linear spaces. Most results in this chapter are known in some form, but scattered among the available literature. The main purpose of this chapter is to collect these results and to determine the algebraic conditions that suffice to give linear algebra over the semifield a familiar feel. For example, under which conditions are varieties cut out by linear polynomials closed under addition and scalar multiplication? The fourth chapter comprises the biggest part of the thesis. The techniques used are a combination of tropical linear algebra and matroid theory. Central objects are the valuated matroids introduced by Andreas Dress and Walter Wenzl. Among other things the chapter contains a classification of functions on a tropical linear space whose cycles are tropical linear subspaces, extending an old result on elementary extensions of matroids by Henry Crapo. It uses Mikhalkin’s concept of a tropical modification to define the morphisms in a category whose objects are all tropical linear spaces. Finally, we determine the structure of an open submonoid of the morphisms from affine 2-space to itself as a polyhedral complex. Finally, the fifth and last chapter is only indirectly related to maps. It studies a certain monoid contained in the tropicalization of the orthogonal group: the monoid that is generated by the distance matrices under tropical matrix multiplication (i.e. where addition is replaced by minimum, and multiplication by addition). This monoid generalizes a monoid that underlies the well-known gossip problem, to a setting where information is transmitted only with a certain degree accuracy. We determine this so-called gossip monoid for matrices up to size 4, and prove that in general it is a polyhedral monoid of dimension equal to that of the orthogonal group.