Semiring Congruences and Tropical Geometry
Semiring Congruences and Tropical Geometry
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半环同余和热带几何
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
Kalina Mincheva
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作者：
Kalina Mincheva
本文的主要动机和启示之一是关于特征几何中几何定义这一尚未解决的问题。这是一个结构上的几何，称为幂等半环，其中1 + 1 = 1。虽然数学家们已经研究半环很多年了，但这些结构只是最近才引起了人们对代数几何，更确切地说是热带几何的兴趣。这是一个特殊的幂等元半环上的几何，即热带半域。此外，半环在数论中也有重要的应用。Connes和C. Consani致力于寻找黎曼假设的新方法。定义了交换半环的素谱。由于理想不再保留它们在半环理论中的特殊作用，这个谱的点对应于某些同余关系，我们称之为素同余。受热带几何的启发，我们工作的主题是研究热带多项式半环的素谱，但这里的许多结果也适用于任何加法幂等半环。我们引入的素同余类表现出与交换环的素理想类似的性质。为了建立一个好的根同余概念，我们证明了半环的所有素数的交可以用某些扭幂公式来刻画。给出了热带半域Rmax、半域Zmax和布尔半域B上多项式半环和Laurent多项式半环的素同余的完整刻画。这些半环的最小素数对应于单项序，它们的交集是确定具有相同牛顿多面体的多项式的同余。我们表明，在这些情况下，每一个素生成的同余的根是一个交叉的素同余的Krull维数为1。利用这一方法，我们证明了本文的一个主要结果，改进了A. Bertram和R.伊斯顿，它可以被看作是一个零斯特兰特多项式。其余的结果都集中在Krull维数的概念。我们证明了对任何幂等半环A，我们有dimA[x] = dimA+ 1。在这种情况下，当我们研究
One of the main motivations and inspirations for this thesis is the still open question of the definition of geometry in characteristic one. This is geometry over a structure, called an idempotent semiring, in which 1 + 1 = 1. While mathematicians have studied semirings for many years, these structures have only recently ignited interest in algebraic geometry, more precisely tropical geometry. This is geometry over a particular idempotent semiring the tropical semifield. Furthermore, semirings have important number theoretic applications which appear in the work of A. Connes and C. Consani which is focused on finding a new approach to the Riemann hypothesis. We define the prime spectrum of a commutative semiring. Since ideals do not retain their distinguished role in the theory of semirings, the points of this spectrum correspond to certain congruence relations, which we call prime congruences. Motivated by tropical geometry, the key theme of our work is to study the prime spectrum of tropical polynomial semirings, but many of the results presented here apply to any additively idempotent semiring as well. The class of prime congruences which we introduce turns out to exhibit some analogous properties to the prime ideals of commutative rings. In order to establish a good notion of radical congruences, we show that the intersection of all primes of a semiring can be characterized by certain twisted power formulas. We give a complete description of prime congruences in the polynomial and Laurent polynomial semirings over the tropical semifield Rmax, the semifield Zmax and the Boolean semifield B. The minimal primes of these semirings correspond to monomial orderings, and their intersection is the congruence that identifies polynomials that have the same Newton polytope. We show that the radical of every finitely generated congruence in each of these cases is an intersection of prime congruences with quotients of Krull dimension 1. Using this setup we prove one of the main results of this thesis we improve on a result of A. Bertram and R. Easton which can be regarded as a Nullstellensatz for tropical polynomials. The remaining results are centered about the concept of Krull dimension. We prove that for any idempotent semiring A we have that dimA[x] = dimA+ 1. In the case when we work over the
影响因子：
2.5
作者：
Jeffrey Giansiracusa;Noah Giansiracusa
通讯作者：
Jeffrey Giansiracusa;Noah Giansiracusa