半环的赋值
Valuations of Semirings
摘要：
我们开发的概念，半环上的估值，以期扩展抽象非奇异曲线和离散赋值环的经典理论，这个一般的代数设置，我们的方法的新颖性在于实施超环产生一个新的定义（\emph{hyperfield valuation}）。特别是，我们分类的评价上的半域$\mathbb{Q}_{max}$（最大加半域的有理数），也评价上的“功能域”$\mathbb{Q}_{max}(T)$（半域的有理函数在$\mathbb{Q}_{max}$），这是平凡的$\mathbb{Q}_{max}$。本文构造并研究了$\mathbb{Q}_{max}(T)$与域上单元素$\mathbb{F}_{1}$和热带投影线$\mathbb{P}^1_{\mathbb{F}_1}$的关系.最后，我们讨论可能的连接到热带曲线和Berkovich的理论的解析空间。
英文摘要：
We develop notions of valuations on a semiring, with a view toward extending the classical theory of abstract nonsingular curves and discrete valuation rings to this general algebraic setting; the novelty of our approach lies in the implementation of hyperrings to yield a new definition (\emph{hyperfield valuation}). In particular, we classify valuations on the semifield $\mathbb{Q}_{max}$ (the max-plus semifield of rational numbers) and also valuations on the `function field' $\mathbb{Q}_{max}(T)$ (the semifield of rational functions over $\mathbb{Q}_{max}$) which are trivial on $\mathbb{Q}_{max}$. We construct and study the abstract curve associated to $\mathbb{Q}_{max}(T)$ in relation to the projective line $\mathbb{P}^1_{\mathbb{F}_1}$ over the field with one element $\mathbb{F}_{1}$ and the tropical projective line. Finally, we discuss possible connections to tropical curves and Berkovich's theory of analytic spaces.
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