Searcharxiv⌕ Search

arXiv subjects

JuAe Song

Publications and source records attributed to JuAe Song.

13 recordsLinked to original sources
arXiv2026

Tropical curves with parallel rays

In the previous works, the rational function semifields of abstract tropical curves were characterized. In this paper, we give a contravariant categorical equivalence between the category of abstract tropical curves with morphisms and the category of semifields over the tropical semifield $\boldsymbol{T}$ characterized above with $\boldsymbol{T}$-algebra homomorphisms. The characterization tells us that the traditional definition of abstract tropical curves has a fatal flaw such that we are never able to deal with parallel rays, unlike the traditional tropical curves, which generally admit them. To address this flaw, we introduce a new notion of abstract tropical curves with parallel rays. Then we define the rational function semifields of these curves and give a characterization of them, and a variant of the categorical equivalence between their categories with a suitable notion of morphisms between these curves. Under the categorical equivalences, we translate several geometric notions for traditional or abstract tropical curves (with parallel rays) into algebraic ones, including weights on edges and the balancing condition.

math.AG↗
arXiv2025

Tropical function fields, finite generation, and faithful tropicalization

Given an algebraic variety defined over a discrete valuation field and a skeleton of its Berkovich analytification, the tropicalization process transforms function field of the variety to a semifield of tropical functions on the skeleton. Our main result offers a purely polyhedral characterization of this semifield: we show that a tropical function is in the image of the tropicalization map if and only if it takes the same slope near infinity along parallel half-lines of the skeleton. This extends a result of Baker and Rabinoff in dimension one to arbitrary dimensions. We use this characterization to establish that this semifield is finitely generated over the semifield of tropical rational numbers, providing a new proof of a recent result by Ducros, Hrushovski, Loeser and Ye in the discrete valued field case. As a second application, we present a new proof of the faithful tropicalization theorem by Gubler, Rabinoff and Werner in the discrete valuation field case. The proof is constructive and provides explicit coordinate functions for the embedding of the skeleton, extending the existing results in dimension one to arbitrary dimensions.

math.AG↗
arXiv2024

A geometric interpretation of Krull dimensions of $\boldsymbol{T}$-algebras

We investigate Krull dimensions of semirings and semifields dealt in tropical geometry. For a congruence $C$ on a tropical Laurent polynomial semiring $\boldsymbol{T}[X_1^{\pm}, \ldots, X_n^{\pm}]$, a finite subset $T$ of $C$ is called a finite congruence tropical basis of $C$ if the congruence variety $\boldsymbol{V}(T)$ associated with $T$ coincides with $\boldsymbol{V}(C)$. For $C$ proper, we prove that the Krull dimension of the quotient semiring $\boldsymbol{T}[X_1^{\pm}, \ldots, X_n^{\pm}] / C$ coincides with the maximum of the dimension of $\boldsymbol{V}(C)$ as a polyhedral complex plus one and that of $\boldsymbol{V}(C_{\boldsymbol{B}})$ when both $C$ and $C_{\boldsymbol{B}}$ have finite congruence tropical bases, respectively. Here $C_{\boldsymbol{B}}$ is the congruence on $\boldsymbol{T}[X_1^{\pm}, \ldots, X_n^{\pm}]$ generated by $\{ (f_{\boldsymbol{B}}, g_{\boldsymbol{B}}) \,|\, (f, g) \in C \}$ and $f_{\boldsymbol{B}}$ is defined as the tropical Laurent polynomial obtained from $f$ by replacing the coefficients of all non $-\infty$ terms of $f$ with the real number zero. With this fact, we also show that rational function semifields of tropical curves that do not consist of only one point have Krull dimension two.

math.AG↗
arXiv2024

Finitely generated congruences on tropical rational function semifields

We prove that the congruence on the tropical rational function semifield in $n$-variables associated with a subset $V$ of $\boldsymbol{R}^n$ is finitely generated if and only if the closure of $V$ is a finite union of $\boldsymbol{R}$-rational polyhedral sets. With this fact, we characterize rational function semifields of tropical curves.

math.AG↗
arXiv2023

Congruences on tropical rational function semifields and tropical curves

We define tropical rational function semifields $\overline{\boldsymbol{T}(X_1, \ldots, X_n)}$ and prove that a tropical curve $\varGamma$ is realized (except for points at infinity) as the congruence variety $V \subset \boldsymbol{R}^n$ associated with a congruence on $\overline{\boldsymbol{T}(X_1, \ldots, X_n)}$ by giving a specific map $\varGamma \to V$. Also, we shed light on the relation between congruences $E$ on $\overline{\boldsymbol{T}(X_1, \ldots, X_n)}$ and congruence varieties associated with them and reveal the quotient semifield $\overline{\boldsymbol{T}(X_1, \ldots, X_n)} / E$ to play the role of coordinate rings that determine isomorphism classes of affine varieties in the classical algebraic geometry.

math.AG↗
arXiv2022

Galois actions for semifield extensions and Galois coverings on tropical curves

For a semifield extension $T /S$, an action of a finite group $G$ on $T$ is Galois if $(1)$ the $G$-invariant subsemifield of $T$ is $S$ and $(2)$ subgroups of $G$ whose invariant semifields coincide are equal. We show that for a finite harmonic morphism between tropical curves $\varphi : \varGamma \to \varGamma^{\prime}$ and an isometric action of a finite group $G$ on $\varGamma$, $\varphi$ is $G$-Galois if and only if the natural action of $G$ on the rational function semifield $\operatorname{Rat}(\varGamma)$ of $\varGamma$ induced by the action of $G$ on $\varGamma$ is Galois for the semifield extension $\operatorname{Rat}(\varGamma) / \varphi^{\ast}(\operatorname{Rat}(\varGamma^{\prime}))$, where $\varphi^{\ast}(\operatorname{Rat}(\varGamma^{\prime}))$ stands for the pull-back of $\operatorname{Rat}(\varGamma^{\prime})$ by $\varphi$.

math.AC↗
arXiv2021

Rational function semifields of tropical curves are finitely generated over the tropical semifield

We prove that the rational function semifield of a tropical curve is finitely generated as a semifield over the tropical semifield $\boldsymbol{T} := ( \boldsymbol{R} \cup \{ - \infty \}, \operatorname{max}, +)$ by giving a specific finite generating set. Also, we show that for a finite harmonic morphism between tropical curves $\varphi : \varGamma \to \varGamma^{\prime}$, the rational function semifield of $\varGamma$ is finitely generated as a $\varphi^{\ast}(\operatorname{Rat}(\varGamma^{\prime}))$-algebra, where $\varphi^{\ast}(\operatorname{Rat}(\varGamma^{\prime}))$ stands for the pull-back of the rational function semifield of $\varGamma^{\prime}$ by $\varphi$.

math.AG↗
arXiv2021

Semiring isomorphisms between rational function semifields of tropical curves induce isomorphisms between tropical curves

We prove that a semiring isomorphism between the rational function semifields of two tropical curves induces an expansive map between those tropical curves. This semiring isomorphism and the expansive map respect zeros and poles of rational functions with their degrees. As a corollary, we show that the automorphism group of a tropical curve is isomorphic to the $\boldsymbol{T}$-algebra automorphism group of its rational function semifield, where $\boldsymbol{T} := (\boldsymbol{R} \cup \{ -\infty \}, \operatorname{max}, +)$ is the tropical semifield. Finally, we describe all semiring automorphisms of rational function semifields of all tropical curves.

math.AG↗
arXiv2021

Upper bounds of orders of automorphism groups of leafless metric graphs

We prove a tropical analogue of the theorem of Hurwitz: a leafless metric graph of genus $g \ge 2$ has at most $12$ automorphisms when $g = 2$; $2^g g!$ automorphisms when $g \ge 3$. These inequalities are optimal; for each genus, we give all metric graphs which have the maximum numbers of automorphisms. The proof is written in terms of graph theory.

math.CO↗
arXiv2019

Galois quotients of metric graphs and invariant linear systems

For a map $φ: \varGamma \rightarrow \varGamma^{\prime}$ between metric graphs and an isometric action on $\varGamma$ by finite group $K$, $φ$ is a $K$-Galois covering on $\varGamma^{\prime}$ if $φ$ is a morphism, the degree of $φ$ coincides with the order of $K$ and $K$ induces a transitive action on every fibre. We prove that for a metric graph $\varGamma$ with an isometric action by finite group $K$, there exists a rational map, from $\varGamma$ to a tropical projective space, which induces a $K$-Galois covering on the image. By using this fact, we also prove that for a hyperelliptic metric graph without one valent points and with genus at least two, the invariant linear system of the hyperelliptic involution $ι$ of the canonical linear system, the complete linear system associated to the canonical divisor, induces an $\langle ι\rangle$-Galois covering on a tree. This is an analogy of the fact that a compact Riemann surface is hyperelliptic if and only if the canonical map, the rational map induced by the canonical linear system, is a double covering on a projective line $\boldsymbol{P}^1$.

math.AG↗