发表机构
Faculty of Mathematics, Kyushu University（九州大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为半环建立同余谱的几何理论，证明一类半环同余谱的拟紧性，并重构热带商半环，应用于热带曲线有理函数半域以恢复曲线、射线平行性及度量。
AI 中文摘要
我们为半环的同余谱发展了一套几何理论。我们将半环的同余谱定义为其上素同余的集合，赋予其Zariski拓扑，并构造一个结构层。对于包括$\oldsymbol{B}$-代数在内的一类半环，我们建立了其同余谱的拟紧性。随后，我们证明了商$K/\oldsymbol{E}(V)$的同余谱的重构结果，其中$V$是欧几里得空间的子集，$K$是热带洛朗多项式半环、热带洛朗多项式函数半环或热带有理函数半域，而$\oldsymbol{E}(V)$是$K$中在$V$上取值一致的元素对所构成的同余。结构层的整体截面恢复出$K/\oldsymbol{E}(V)$，且相应的代数范畴与几何范畴是反变等价的。对于$\oldsymbol{R}$-有理多面体集$V$的有限并，我们完全分类了$K / \oldsymbol{E}(V)$上的素同余。作为应用，我们研究了具有平行射线的热带曲线的有理函数半域。我们证明它们的同余谱恢复出底层曲线以及射线的平行性和它们的内在度量。
英文摘要
We develop a geometric theory of congruence spectra for semirings. We define the congruence spectrum for a semiring as the set of prime congruences on it, equip it with the Zariski topology, and construct a structure sheaf. For a class of semirings including $\boldsymbol{B}$-algebras, we establish quasi-compactness of their congruence spectra. We then prove reconstruction results for congruence spectra of quotients $K/\boldsymbol{E}(V)$, where $V$ is a subset of Euclidean space, $K$ is a tropical Laurent polynomial semiring, a tropical Laurent polynomial function semiring, or a tropical rational function semifield, and $\boldsymbol{E}(V)$ is the congruence of pairs of elements of $K$ that agree on $V$. The global sections of the structure sheaf recover $K/\boldsymbol{E}(V)$, and the corresponding algebraic and geometric categories are contravariantly equivalent. For finite unions of $\boldsymbol{R}$-rational polyhedral sets $V$, we completely classify the prime congruences on $K / \boldsymbol{E}(V)$. As an application, we study rational function semifields of tropical curves with parallel rays. We show that their congruence spectra recover the underlying curves together with the parallelism of rays and their intrinsic metrics.
Comments17 pages
