# Finitely generated congruences on tropical rational function semifields

Song JuAe  Department of Mathematics, Graduate School of Science, Kyoto
University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan.
<EMAIL_ADDRESS>
###### Abstract.

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.

###### Key words and phrases:

tropical rational function semifields, congruences, congruence varieties,
$\boldsymbol{R}$-rational polyhedral sets, tropical curves

###### 2020 Mathematics Subject Classification:

14T10, 14T15, 14T20, 14T25, 15A80

##  1\. Introduction

The name of tropical geometry was given in [23]. It is an algebraic geometry
over the tropical semifield
$\boldsymbol{T}:=(\boldsymbol{R}\cup\\{-\infty\\},\oplus:=\operatorname{max},\odot:=+)$
and has many sources; e.g., Bergman fans ([1]), amoebas ([5]), tropical
algebra (introduced by Imre Simon, cf. [22]) and patchworkings (cf. [24]). In
the last two decades, it has been revealed that tropical geometry has many
applications in other areas. For example, Mikhalkin’s correspondence Principle
([21]) states that the Gromov–Witten numbers can be found tropically. Another
proof of Brill–Noether theorem was given in [4]. Tropical polynomials are used
to describe the earliest finishing times of project networks in [19].

By such backgrounds, it is expected that tropical geometry itself has an
interest and rich structure. To construct algebraic foundation for tropical
geometry, Bertram and Easton ([2]) and Joó and Mincheva ([17]) proved a
tropical Nullstellensatz for congruences on tropical polynomial semirings
$\boldsymbol{T}[X_{1},\ldots,X_{n}]$. As in the classical algebraic geometry,
corresponding to a congruence $E$ on $\boldsymbol{T}[X_{1},\ldots,X_{n}]$, the
geometric object, called a congruence variety,
$\boldsymbol{V}(E)_{0}\subset\boldsymbol{T}^{n}$ is defined, and a subset $V$
of $\boldsymbol{T}^{n}$ defines the congruence $\boldsymbol{E}(V)_{0}$ on
$\boldsymbol{T}[X_{1},\ldots,X_{n}]$ (see Subsection 2.4 for more details).
With these concepts, the tropical Nullstellensatz gives a characterization of
$\boldsymbol{E}(\boldsymbol{V}(E)_{0})_{0}$ when $E$ is finitely generated as
a congruence, i.e., states that $\boldsymbol{E}(\boldsymbol{V}(E)_{0})_{0}$
coincides with the congruence $\widehat{E}$ defined from $E$ with specific
conditions. There is a problem that the tropical Nullstellensatz does not
ensure that $\widehat{E}$ is also finitely generated as a congruence.

The current paper is devoted to solving this problem when $E$ is a finitely
generated congruence on the tropical rational function semifield
$\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$. More precisely, we prove the
following theorem:

######  Theorem 1.1.

Let $V$ be a subset of $\boldsymbol{R}^{n}$. Then $\boldsymbol{E}(V)$ is
finitely generated if and only if the closure (in the usual sense)
$\overline{V}$ of $V$ in $\boldsymbol{R}^{n}$ is a finite union of
$\boldsymbol{R}$-rational polyhedral sets.

Here $\boldsymbol{E}(V)$ denotes the congruence on
$\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$ associated with $V$. In the
proof, we obtain a specific generating set of $\boldsymbol{E}(V)$ consisting
of only one element explicitly. This theorem has the following several
corollaries:

######  Corollary 1.2.

For a congruence $E$ on $\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$, if
$E$ is finitely generated as a congruence, then so is
$\boldsymbol{E}(\boldsymbol{V}(E))$.

Here $\boldsymbol{V}(E)$ is the congruence variety associated with $E$ and is
in $\boldsymbol{R}^{n}$.

######  Corollary 1.3 (Corollary 3.12).

Let $V$ (resp. $W$) be a subset of $\boldsymbol{R}^{n}$ (resp.
$\boldsymbol{R}^{m}$). When the quotient $\boldsymbol{T}$-algebra
$\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}/\boldsymbol{E}(V)$ is
isomorphic to
$\overline{\boldsymbol{T}(Y_{1},\ldots,Y_{m})}/\boldsymbol{E}(W)$ as a
$\boldsymbol{T}$-algebra, $\boldsymbol{E}(V)$ is finitely generated as a
congruence if and only if so is $\boldsymbol{E}(W)$.

######  Corollary 1.4 (cf. [18, Proposition 3.9(i)]).

Let $V$ be a finite union of $\boldsymbol{R}$-rational polyhedral sets in
$\boldsymbol{R}^{n}$. Then there exist
$f_{1},f_{2}\in\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}$ such that
$\boldsymbol{E}(V)_{1}=\\{(g,h)\in\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}\times\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}\,|\,\exists
k\in\boldsymbol{Z}_{>0}:(g\odot f_{2}^{\odot k},h\odot f_{2}^{\odot
k})\in\langle(f_{1},f_{2})\rangle\\}$ and $f_{1}(x)\geq f_{2}(x)$ for any
$x\in\boldsymbol{T}^{n}$. In particular, if $f_{2}$ is taken as a constant
tropical monomial of a real number, then
$\boldsymbol{E}(V)_{1}=\langle(f_{1},f_{2})\rangle$ holds.

Here $\boldsymbol{E}(V)_{1}$ is the congruence on the tropical polynomial
function semiring $\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}$ associated
with $V$ and $\langle(f_{1},f_{2})\rangle$ stands for the congruence on
$\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}$ generated by $(f_{1},f_{2})$.

Theorem 1.1 also gives a characterization of rational function semifields
$\operatorname{Rat}(\varGamma)$ of tropical curves $\varGamma$ (Corollary
3.19). Note here that (parts of) [13], [14], [15] and [16] were devoted to
obtain this characterization. With this characterization, we know that if the
rational map $\phi$ to $\boldsymbol{R}^{n}$ induced by the complete linear
system associated with an effective divisor on $\varGamma$ is injective, then
$\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}/\boldsymbol{E}(\operatorname{Im}(\phi))$
is isomorphic to $\operatorname{Rat}(\varGamma)$ as a $\boldsymbol{T}$-algebra
(and hence as a semifield over $\boldsymbol{T}$) (Corollary 3.21), where
$\operatorname{Im}(\phi)$ stands for the image of $\phi$. This means that
$\phi$ has all imformation of $\operatorname{Rat}(\varGamma)$ as a
$\boldsymbol{T}$-algebra and all geometric imformation $\varGamma$ as a
tropical curve by [16, Corollary 1.2], intuitively, $\phi$ is a “good”
embedding.

Theorem 1.1 is a tropical analogue of the Noether’s property. However there
exists no conclusive definition of (abstract) tropical varieties, Theorem 1.1
ensures that arbitrary “tropical” geometric object $V$ in $\boldsymbol{R}^{n}$
in the broadest sense is defined by a finite system of equalities of tropical
rational functions and defines the finitely generated semifield
$\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}/\boldsymbol{E}(V)$ over
$\boldsymbol{T}$ (cf. the Structure Theorem for Tropical Geometry [20, Theorem
3.3.5]). We expect the current paper to become a new step toward the
construction of algebraic foundation for tropical geometry.

Note here that a part of Theorem 1.1 is indicated in the introduction of [9].

The rest of this paper is organized as follows. In Section 2, we give the
definitions of polyhedral cones and polyhedral sets, semirings, algebras and
semifields, congruences, tropical rational function semifields and congruence
varieties, tropical curves, rational functions, morphisms between tropical
curves, and linear systems. In Section 3, which is our main section, we prove
the above assertions.

## Acknowledgements

The author thanks her supervisor Masanori Kobayashi, Kazuhiko Yamaki, Yasuhito
Nakajima, Keita Goto, Masao Oi, Yuki Koto, Taro Sogabe and Takaaki Ito for
their helpful comments and discussions.

##  2\. Preliminaries

In this section, we recall several definitions which we need later. We refer
to [8] (resp. [20]) for an introduction to the theory of semirings (resp.
tropical geometry and polyhedral geometry) and employ definitions in [2], [15]
and [17] (resp. [12], [13] and [20]) with slight modifications partially.
Also, we employ Chan’s definition of morphisms between tropical curves in [3].

###  2.1. Polyhedral cones and polyhedral sets

For any subset $S$ of $\boldsymbol{R}^{n}$, the polar cone $S^{\ast}$ of $S$
is defined by the set consisting of all points $x$ in $\boldsymbol{R}^{n}$
such that $x\cdot y\leq 0$ holds for any $y\in S$, where $\cdot$ denotes the
standard inner product. It is well-known that $S^{\ast}$ is a closed convex
cone, i.e., $S^{\ast}$ is closed and for any $x,y\in S^{\ast}$ and
$r\in\boldsymbol{R}_{\geq 0}$, both the segment $\overline{xy}$ and the point
$rx$ are in $S^{\ast}$, and $(K^{\ast})^{\ast}=K$ holds for any closed convex
cone $K$. The following theorem is well-known and one of the keys to our main
theorem:

######  Theorem 2.1.

Let $K$ be a nonempty closed convex cone. For any point $x$ of
$\boldsymbol{R}^{n}$, there exist unique points $y$ of $K$ and $z$ of
$K^{\ast}$ satisfying $x=y+z$ and $y\cdot z=0$.

Note that in Theorem 2.1, $\operatorname{dist}(x,y)=\operatorname{dist}(x,K)$
holds, where $\operatorname{dist}(x,y)$ (resp. $\operatorname{dist}(x,K)$)
denotes the distance between $x$ and $y$ (resp. $K$). For any subset $S$ of
$\boldsymbol{R}^{n}$, let $\operatorname{Cone}(S)$ stand for the convex cone
hull of $S$, i.e., the set of all points $x$ of $\boldsymbol{R}^{n}$ such that
there exist a finite number of nonnegative numbers $t_{1},\ldots,t_{k}\geq 0$
and $k$ elements $s_{1},\ldots,s_{k}$ of $S$ satisfying
$x=t_{1}s_{1}+\cdots+t_{k}s_{k}$. A polyhedral cone is the convex cone hull of
a finite number of points of $\boldsymbol{R}^{n}$, which is closed. If $K$ is
a polyhedral cone, then so is $K^{\ast}$. Since $(K^{\ast})^{\ast}=K$, a
polyhedral cone is the solution set of a system of (a finite number of)
homogeneous linear inequalities. For $a\in\boldsymbol{R}^{n}$ and
$b\in\boldsymbol{R}$, if $K$ is contained in
$\\{x\in\boldsymbol{R}^{n}\,|\,a\cdot x+b\geq 0\\}$, then
$K\cap\\{x\in\boldsymbol{R}^{n}\,|\,a\cdot x+b=0\\}$ is called a face of $K$.
A proper face of $K$ is a face other than $\varnothing,K$. For two subsets
$S_{1},S_{2}$ of $\boldsymbol{R}^{n}$, let $S_{1}+S_{2}$ denote the Minkowski
sum of $S_{1}$ and $S_{2}$, i.e., the set $\\{s_{1}+s_{2}\,|\,s_{1}\in
S_{1},s_{2}\in S_{2}\\}$. For a subset $S$ of $\boldsymbol{R}^{n}$, let
$\operatorname{Int}(S)$ and $\partial(S)$ stand for the relative interior of
$S$ and the boundary of $S$, respectively. Theorem 2.1 gives a decomposition
of $\boldsymbol{R}^{n}$ in the case that $K$ is a nonempty polyhedral cone:

######  Corollary 2.2.

For a finite number of points $x_{1},\ldots,x_{m}$ of
$\boldsymbol{R}^{n}\setminus\\{(0,\ldots,0)\\}$, let $K$ be the polyhedral
cone $\\{y\in\boldsymbol{R}^{n}\,|\,\forall i,x_{i}\cdot y\leq 0\\}$. If $K$
is nonempty, then

$\displaystyle\boldsymbol{R}^{n}$ $\displaystyle=K+K^{\ast}$
$\displaystyle=K+\operatorname{Cone}\\{x_{1},\ldots,x_{m}\\}$
$\displaystyle=\operatorname{Int}(K)\sqcup\bigsqcup_{H:\text{a proper face of
}K}(\operatorname{Int}(H)+\operatorname{Cone}\\{x_{i}\,|\,x_{i}\perp H\\})$
$\displaystyle=\operatorname{Int}(K)\sqcup\bigsqcup_{y\in\partial(K)}(\\{y\\}+\operatorname{Cone}\\{x_{i}\,|\,x_{i}\cdot
y=0\\}).$

We extend Corollary 2.2 to the case of polyhedral sets. Note that a polyhedral
set is the solution set of a system of (a finite number of) linear
inequalities. A polyhedral set is $\boldsymbol{R}$-rational if all
coefficients of its system of (a finite number of) linear inequalities except
for constant terms are rational.

######  Corollary 2.3.

For a finite number of points $x_{1},\ldots,x_{m}$ of
$\boldsymbol{R}^{n}\setminus\\{(0,\ldots,0)\\}$ and real numbers
$a_{1},\ldots,a_{m}$, let $K$ be the polyhedral set
$\\{y\in\boldsymbol{R}^{n}\,|\,\forall i,x_{i}\cdot y+a_{i}\leq 0\\}$. If $K$
is nonempty, then

$\displaystyle\boldsymbol{R}^{n}=\operatorname{Int}(K)\sqcup\bigsqcup_{y\in\partial(K)}(\\{y\\}+\operatorname{Cone}\\{x_{i}\,|\,x_{i}\cdot
y+a_{i}=0\\}).$

###### Proof.

It is enough to show that for any $z\in\boldsymbol{R}^{n}$ and the unique
point $w$ of $K$ such that $\operatorname{dist}(z,w)=\operatorname{dist}(z,K)$
holds, $z$ is in $\\{w\\}+\operatorname{Cone}\\{x_{i}\,|\,x_{i}\cdot
w+a_{i}=0\\}$. We use the induction on $m$. It is true for $m=1$. We assume
that it is true for $m=k$. We consider the case of $m=k+1$. For any
$z\in\boldsymbol{R}^{n}$, let $w$ be the unique point of $K$ such that
$\operatorname{dist}(z,w)=\operatorname{dist}(z,K)$ holds. Assume that the
cardinality of $\\{i\,|\,x_{i}\cdot w+a_{i}=0\\}$ is less than $k+1$, say
$k+1\not\in\\{i\,|\,x_{i}\cdot w+a_{i}=0\\}$. Then $w$ is a point of the
polyhedral set $L:=\\{y\in\boldsymbol{R}^{n}\,|\,i=1,\ldots,k,x_{i}\cdot
y+a_{i}\leq 0\\}$ and
$\operatorname{dist}(z,w)=\operatorname{dist}(z,K)\geq\operatorname{dist}(z,L)$
and $x_{k+1}\cdot w+a_{k+1}<0$. There exists a unique point $w^{\prime}$ of
$L$ such that $\operatorname{dist}(z,w^{\prime})=\operatorname{dist}(z,L)$. If
$w\not=w^{\prime}$, then
$\operatorname{dist}(z,w)>\operatorname{dist}(z,w^{\prime})$. In this case, if
$x_{k+1}\cdot w^{\prime}+a_{k+1}\leq 0$, then $w^{\prime}\in K$, and hence
$\operatorname{dist}(z,w^{\prime})\geq\operatorname{dist}(z,K)=\operatorname{dist}(z,w)$,
which is a contradiction. Hence we have $x_{k+1}\cdot w^{\prime}+a_{k+1}>0$.
Since both $w$ and $w^{\prime}$ are in $L$ and $L$ is convex, the segment
$\overline{ww^{\prime}}$ is contained in $L$. Thus there exists a unique point
$w^{\prime\prime}$ of $\overline{ww^{\prime}}$ such that $x_{k+1}\cdot
w^{\prime\prime}+a_{k+1}=0$. This $w^{\prime\prime}$ is a point of $K$. As
$\operatorname{dist}(z,w^{\prime})<\operatorname{dist}(z,w)$, we have
$\operatorname{dist}(z,w^{\prime\prime})<\operatorname{dist}(z,w)=\operatorname{dist}(z,K)$,
which is a contradiction. Therefore $w^{\prime}$ must coincide with $w$. By
the assumption of the induction, we have
$z\in\\{w\\}+\operatorname{Cone}\\{x_{i}\,|\,x_{i}\cdot
w+a_{i}=0,i\not=k+1\\}\subset\\{w\\}+\operatorname{Cone}\\{x_{i}\,|\,x_{i}\cdot
w+a_{i}=0\\}$. Assume that the cardinality of $\\{i\,|\,x_{i}\cdot
w+a_{i}=0\\}$ is $k+1$. Since $m=k+1$, we have
$K=\\{w\\}+\\{y\in\boldsymbol{R}^{n}\,|\,\forall i,x_{i}\cdot y\leq 0\\}$. By
Corollary 2.2, $z-w\in\operatorname{Cone}\\{x_{1},\ldots,x_{k+1}\\}$, and thus
$z\in\\{w\\}+\operatorname{Cone}\\{x_{1},\ldots,x_{k+1}\\}$. ∎

A convex cone in $\boldsymbol{R}^{n}$ is strongly convex if it contains no
lines through the origin $O$ of $\boldsymbol{R}^{n}$. For a polyhedral cone
$K$ in $\boldsymbol{R}^{n}$, it is well-known that $K$ is strongly convex if
and only if $K^{\ast}$ is $n$-dimensional and that if $K$ is not strongly
convex, then there exist $\\{i_{1},\ldots,i_{s}\\}\subset\\{1,\ldots,m\\}$ and
$c_{i_{1}},\ldots,c_{i_{s}}>0$ such that $\sum_{j=1}^{s}c_{i_{j}}x_{i_{j}}=O$
holds.

######  Lemma 2.4.

For a finite number of points $x_{1},\ldots,x_{m}$ of
$\boldsymbol{R}^{n}\setminus\\{(0,\ldots,0)\\}$, there exists $k>0$ such that
$\sqrt{y\cdot y}\leq k\operatorname{max}\\{x_{i}\cdot y\,|\,i=1,\ldots,m\\}$
holds for any point $y$ of $\operatorname{Cone}\\{x_{1},\ldots,x_{m}\\}$.

###### Proof.

Let $K$ be the polyhedral cone $\operatorname{Cone}\\{x_{1},\ldots,x_{m}\\}$
and $d:K\to\boldsymbol{R}_{\geq 0};y\mapsto\sqrt{y\cdot y}$ and
$d^{\prime}:K\to\boldsymbol{R}_{\geq
0};y\mapsto\operatorname{max}\\{x_{i}\cdot y\,|\,i=1,\ldots,m\\}$. Note that
$d^{\prime}(y)$ must be nonnegative for any $y\in K$. In fact, if $y\in K$
satisfies $x_{i}\cdot y<0$ for $i=1,\ldots,m$, then $y\in K^{\ast}$, and hence
$y\in K\cap K^{\ast}$ and $y$ is not $O$. However, by Theorem 2.1, $K\cap
K^{\ast}$ is consiting only of $O$, which is a contradiction.

Assume that $K$ is strongly convex. Let $y$ be a point of
$\operatorname{Int}(K)$. By the assumption above,
$L:=K\cap\partial(\\{y\\}+K^{\ast})$ is a nonempty compact subset of
$\boldsymbol{R}^{n}$. Since the functions $d$ and $d^{\prime}$ are continuous,
there exist the maximum value of $d$ on $L$ and the minimum value $D^{\prime}$
of $d^{\prime}$ on $L$. As $y\in\operatorname{Int}(K)$, $L$ does not contain
$O$, and hence $D^{\prime}$ is positive. Hence there exists $k>0$ such that
$d(z)\leq kd^{\prime}(z)$ holds for any $z\in L$. For $y^{\prime}:=ty$ with
$t>0$ and $z^{\prime}\in K\cap\partial(\\{y^{\prime}\\}+K^{\ast})$, since
$\frac{1}{t}z^{\prime}\in L$, we have
$d(z^{\prime})=td\left(\frac{1}{t}z^{\prime}\right)\leq
tkd^{\prime}\left(\frac{1}{t}z^{\prime}\right)=kd^{\prime}(z^{\prime})$. As
$d(O)=d^{\prime}(O)=0$, we have the conclusion in this case.

Assume that $K$ is not strongly convex. Then there exist
$\\{i_{1},\ldots,i_{s}\\}\subset\\{1,\ldots,m\\}$ and
$c_{i_{1}},\ldots,c_{i_{s}}>0$ such that $\sum_{j=1}^{s}c_{i_{j}}x_{i_{j}}=O$
holds. It is clearly true that
$K\supset\bigcup_{j=1}^{s}\operatorname{Cone}\\{x_{l}\,|\,l\not=i_{j}\\}$.
Conversely, for any $y\in K$, there exist $t_{1},\ldots,t_{m}\geq 0$ such that
$y=\sum_{i=1}^{m}t_{i}x_{i}$. Without loss of generality, we can assume that
$i_{j}=j$, $c_{i_{j}}=1$ for $j=1,\ldots,s$ and
$\operatorname{min}\\{t_{i_{j}}\,|\,j=1,\ldots,s\\}=t_{i_{1}}$. Then we have
$y=\sum_{i=1}^{m}t_{i}x_{i}=\sum_{i=1}^{s}(t_{i}-t_{1})x_{i}+\sum_{i=s+1}^{m}t_{i}x_{i}\in\operatorname{Cone}\\{x_{2},\ldots,x_{m}\\}$.
Thus we have
$K=\bigcup_{j=1}^{s}\operatorname{Cone}\\{x_{l}\,|\,l\not=i_{j}\\}$. By
repeating this process at most finitely many times, $K$ is subdivided into
finitely many strongly convex polyhedral cones $H_{j}$ generated by subsets
$X_{H_{j}}$ of $\\{x_{1},\ldots,x_{m}\\}$ for $j=1,\ldots,p$. By the previous
argument, there exists $k_{H_{j}}>0$ such that $d(y)\leq
k_{H_{j}}\operatorname{max}\\{x_{i}\cdot y\,|\,x_{i}\in X_{H_{j}}\\}$ holds
for any point $y$ of $H_{j}$. Since $\operatorname{max}\\{x_{i}\cdot
y\,|\,x_{i}\in X_{H_{j}}\\}\leq\operatorname{max}\\{x_{i}\cdot
y\,|\,i=1,\ldots,m\\}$, for
$k:=\operatorname{max}\\{k_{H_{j}}\,|\,j=1,\ldots,p\\}$, the inequality $d\leq
kd^{\prime}$ holds on $K$, which completes the proof. ∎

A vector $\boldsymbol{v}\in\boldsymbol{R}^{n}$ is primitive if all its
components are integers and their greatest common divisor is one. When
$\boldsymbol{v}$ is primitive, for $\lambda\geq 0$, the lattice length of
$\lambda\boldsymbol{v}$ is defined as $\lambda$ (cf. [12, Subsection 2.1]).

###  2.2. Semirings, algebras and semifields

In this paper, a semiring $S$ is a commutative semiring with the absorbing
neutral element $0_{S}$ for addition $+$ and the identity $1_{S}$ for
multiplication $\cdot$. If every nonzero element of a semiring $S$ is
multiplicatively invertible and $0_{S}\not=1_{S}$, then $S$ is called a
semifield.

A map $\varphi:S_{1}\to S_{2}$ between semirings is a semiring homomorphism if
for any $x,y\in S_{1}$,

$\displaystyle\varphi(x+y)=\varphi(x)+\varphi(y),\ \varphi(x\cdot
y)=\varphi(x)\cdot\varphi(y),\ \varphi(0)=0,\ \text{and}\ \varphi(1)=1.$

Given a semiring homomorphism $\psi:S_{1}\to S_{2}$, if both $S_{1}$ and
$S_{2}$ are semifield and $\psi$ is injective, then $S_{2}$ is a semifield
over $S_{1}$.

Given a semiring homomorphism $\varphi:S_{1}\to S_{2}$, we call the pair
$(S_{2},\varphi)$ (for short, $S_{2}$) a $S_{1}$-algebra. For a semiring
$S_{1}$, a map $\psi:(S_{2},\varphi)\to(S_{2}^{\prime},\varphi^{\prime})$
between $S_{1}$-algebras is a $S_{1}$-algebra homomorphism if $\psi$ is a
semiring homomorphism and $\varphi^{\prime}=\psi\circ\varphi$. When there is
no confusion, we write $\psi:S_{2}\to S_{2}^{\prime}$ simply. A bijective
$S_{1}$-algebra homomorphism $S_{2}\to S_{2}^{\prime}$ is a $S_{1}$-algebra
isomorphism. Then $S_{2}$ and $S_{2}^{\prime}$ are said to be isomorphic.

The set $\boldsymbol{T}:=\boldsymbol{R}\cup\\{-\infty\\}$ with two tropical
operations:

$\displaystyle a\oplus b:=\operatorname{max}\\{a,b\\}\quad\text{and}\quad
a\odot b:=a+b,$

where $a,b\in\boldsymbol{T}$ and $a+b$ stands for the usual sum of $a$ and
$b$, becomes a semifield. Here, for any $a\in\boldsymbol{T}$, we handle
$-\infty$ as follows:

$\displaystyle a\oplus(-\infty)=(-\infty)\oplus a=a\quad\text{and}\quad
a\odot(-\infty)=(-\infty)\odot a=-\infty.$

This triple $(\boldsymbol{T},\odot,\oplus)$ is called the tropical semifield.
The subset $\boldsymbol{B}:=\\{0,-\infty\\}$ of $\boldsymbol{T}$ becomes a
semifield with tropical operations of $\boldsymbol{T}$ and is called the
boolian semifield. A $\boldsymbol{B}$-algebra $S$ is said to be cancellative
if whenever $x\cdot y=x\cdot z$ for some $x,y,z\in S$, then either $x=0_{S}$
or $y=z$. If $S$ is cancellative, then we can define the semifield $Q(S)$ of
fractions as within the case of integral domains. In this case, the map $S\to
Q(S);x\mapsto x/1_{S}$ becomes an injective $\boldsymbol{B}$-algebra
homomorphism. A $\boldsymbol{B}$-algebra $S$ has a natural partial order,
i.e., $x\geq y$ if and only if $x+y=x$ for $x,y\in S$.

The tropical polynomials are defined in the usual way and the set of all
tropical polynomials in $n$-variables is denoted by
$\boldsymbol{T}[X_{1},\ldots,X_{n}]$. It becomes a semiring with two tropical
operations and is called the tropical polynomial semiring. Similarly, we
define the tropical Laurent polynomials and the tropical Laurent polynomial
semiring $\boldsymbol{T}\left[X_{1}^{\pm 1},\ldots,X_{n}^{\pm 1}\right]$. By
[15, Example 2.1], we know that tropical (Laurent) polynomial semirings are
not cancellative.

For a commutative monoid $(M,+)$ is a $\boldsymbol{T}$-module if $(M,+)$ is
equipped with a map $\mu:\boldsymbol{T}\times M\to M$ satisfying

$(1)$ $\mu(t,m_{1}+m_{2})=\mu(t,m_{1})+\mu(t,m_{2})$,

$(2)$ $\mu(t_{1}\oplus t_{2},m)=\mu(t_{1},m)+\mu(t_{2},m)$,

$(3)$ $\mu(t_{1}\odot t_{2},m)=\mu(t_{1},\mu(t_{2},m))$,

$(4)$ $\mu(0,m)=m$, and

$(5)$ $\mu(-\infty,m)=0_{M}$,

where $t,t_{1},t_{2}\in\boldsymbol{T}$, $m,m_{1},m_{2}\in M$ and $0_{M}$
stands for the unity of $M$ (see [6, Subsection 2.1] for more details).

###  2.3. Congruences

A congruence $E$ on a semiring $S$ is a subset of $S^{2}=S\times S$ satisfying

$(1)$ for any $x\in S$, $(x,x)\in E$,

$(2)$ if $(x,y)\in E$, then $(y,x)\in E$,

$(3)$ if $(x,y)\in E$ and $(y,z)\in E$, then $(x,z)\in E$,

$(4)$ if $(x,y)\in E$ and $(z,w)\in E$, then $(x+z,y+w)\in E$, and

$(5)$ if $(x,y)\in E$ and $(z,w)\in E$, then $(x\cdot z,y\cdot w)\in E$.

The diagonal set of $S^{2}$ is denoted by $\Delta$ and called the trivial
congruence on $S$. It is the unique smallest congruence on $S$. The set
$S^{2}$ becomes a semiring with the operations of $S$ and is a congruence on
$S$. This is called the improper congruence on $S$. Congruences other than the
improper congruence are said to be proper. A congruence $E$ on $S$ is finitely
generated as a congruence if there exist
$(x_{1},y_{1}),\ldots,(x_{m},y_{m})\in E$ such that the smallest congruence
$\langle(x_{1},y_{1}),\ldots,(x_{m},y_{m})\rangle_{S}=\langle(x_{1},y_{1}),\ldots,(x_{m},y_{m})\rangle$
on $S$ containing all $(x_{1},y_{1}),\ldots,(x_{m},y_{m})$ is $E$. Quotients
by congruences can be considered in the usual sense and the quotient semiring
of $S$ by the congruence $E$ is denoted by $S/E$. Then the natural surjection
$\pi_{E}:S\twoheadrightarrow S/E$ is a semiring homomorphism.

For a semiring homomorphism $\psi:S_{1}\to S_{2}$, the kernel congruence
$\operatorname{Ker}(\psi)$ of $\psi$ is the congruence $\\{(x,y)\in
S_{1}^{2}\,|\,\psi(x)=\psi(y)\\}$. For semirings and congruences on them, the
fundamental homomorphism theorem holds ([6, Proposition 2.4.4]). Then, for the
above $\pi_{E}$, we have $\operatorname{Ker}(\pi_{E})=E$.

###  2.4. Tropical rational function semifields and congruence varieties

For $V\subset\boldsymbol{T}^{n}$, the set
$\boldsymbol{E}(V)_{0}:=\\{(f,g)\in\boldsymbol{T}[X_{1},\ldots,X_{n}]^{2}\,|\,\forall
x\in V,f(x)=g(x)\\}$ is a congruence on $\boldsymbol{T}[X_{1},\ldots,X_{n}]$.
The semiring
$\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}:=\boldsymbol{T}[X_{1},\ldots,X_{n}]/\boldsymbol{E}(\boldsymbol{T}^{n})_{0}$
is cancellative by [2, Theorem 1] and [17, Proposition 5.5 and Theorem
4.14(v)]. We call it the tropical polynomial function semiring. We call its
semifield of fractions the tropical rational function semifield and write it
$\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$. In what follows, by abuse of
notation, we write the image of $X_{i}$ in
$\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$ as $X_{i}$. For a subset $E$
of $\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}^{2}$, we define
$\boldsymbol{V}(E):=\\{x\in\boldsymbol{R}^{n}\,|\,\forall(f,g)\in
E,f(x)=g(x)\\}$ and call it the congruence variety associated with $E$. By
defining that subsets of $\boldsymbol{R}^{n}$ are closed if they are of the
form of congruence varieties, a topology on $\boldsymbol{R}^{n}$ is determined
and coincides with the Euclidean topology on $\boldsymbol{R}^{n}$ by [15,
Proposition 3.9]. We also prepare the definition
$\boldsymbol{V}(F)_{0}:=\\{x\in\boldsymbol{T}^{n}\,|\,\forall(f,g)\in
F,f(x)=g(x)\\}$ for a subset $F$ of $\boldsymbol{T}[X_{1},\ldots,X_{n}]^{2}$
and the congruence
$\boldsymbol{E}(V)_{1}:=\\{(f,g)\in\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}^{2}\,|\,\forall
x\in V,f(x)=g(x)\\}$ on $\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}$
associated with $V\subset\boldsymbol{T}^{n}$.

###  2.5. Tropical curves

In this paper, a graph is an unweighted, undirected, finite, connected
nonempty multigraph that may have loops. For a graph $G$, the set of vertices
is denoted by $V(G)$ and the set of edges by $E(G)$. A vertex $v$ of $G$ is a
leaf end if $v$ is incident to only one edge and this edge is not a loop. A
leaf edge is an edge of $G$ incident to a leaf end.

A tropical curve is the topological space associated with the pair $(G,l)$ of
a graph $G$ and a function $l:E(G)\to{\boldsymbol{R}}_{>0}\cup\\{\infty\\}$,
where $l$ can take the value $\infty$ only on leaf edges, by identifying each
edge $e$ of $G$ with the closed interval $[0,l(e)]$. The interval $[0,\infty]$
is the one-point compactification of the interval $[0,\infty)$. We regard
$[0,\infty]$ not just as a topological space but as an extended metric space.
The distance between $\infty$ and any other point is infinite. When
$l(e)=\infty$, the leaf end of $e$ must be identified with $\infty$. If
$E(G)=\\{e\\}$ and $l(e)=\infty$, then we can identify either leaf ends of $e$
with $\infty$. When a tropical curve $\varGamma$ is obtained from $(G,l)$, the
pair $(G,l)$ is called a model for $\varGamma$. There are many possible models
for $\varGamma$. We frequently identify a vertex (resp. an edge) of $G$ with
the corresponding point (resp. the corresponding closed subset) of
$\varGamma$. A model $(G,l)$ is loopless if $G$ is loopless. For a point $x$
of a tropical curve $\varGamma$, if $x$ is identified with $\infty$, then $x$
is called a point at infinity, else, $x$ is called a finite point. Let
$\varGamma_{\infty}$ denote the set of all points at infinity of $\varGamma$.
If $x$ is a finite point, then the valency $\operatorname{val}(x)$ is the
number of connected components of $U\setminus\\{x\\}$ with any sufficiently
small connected neighborhood $U$ of $x$; if $x$ is a point at infinity, then
$\operatorname{val}(x):=1$. The genus $g(\varGamma)$ of $\varGamma$ is the
first Betti number of $\varGamma$, which coincides with $\\#E(G)-\\#V(G)+1$
for any model $(G,l)$ for $\varGamma$. If $g(\varGamma)=0$, then $\varGamma$
is a tree. The word “an edge of $\varGamma$” means an edge of $G$ for some
model $(G,l)$ for $\varGamma$. A subgraph of a tropical curve is a closed
subset of the tropical curve with a finite number of connected components.

###  2.6. Rational functions

Let $\varGamma$ be a tropical curve. A continuous map
$f:\varGamma\to\boldsymbol{R}\cup\\{\pm\infty\\}$ is a rational function on
$\varGamma$ if $f$ is a constant function of $-\infty$ or a piecewise affine
function with integer slopes, with a finite number of pieces and that can take
the value $\pm\infty$ at only points at infinity. For a finite point $x$ of
$\varGamma$ and a rational function $f$ on $\varGamma$ other than $-\infty$,
for one outgoing direction at $x$, the outgoing slope of $f$ at $x$ is
$\frac{f(y)-f(x)}{\operatorname{dist}(x,y)}$ with a finite point $y$ in a
sufficiently small neighborhood of $x$ in the dirextion. If $x$ is a point at
infinity, then we regard the outgoing slope of $f$ at $x$ as the slope of $f$
from $y$ to $x$ times minus one, where $y$ is a finite point on the leaf edge
incident to $x$ such that $f$ has a constant slope on the interval $(y,x)$. In
both cases, the definition of outgoing slope of $f$ at $x$ in the direction is
independent of the choice of $y$. Note when $\varGamma$ is a singleton, i.e.,
consists of only one point, for any rational function $f$ other than
$-\infty$, we define $\operatorname{ord}_{x}(f):=0$. Let
$\operatorname{ord}_{x}(f)$ denote the sum of outgoing slopes of $f$ at $x$.
Let $\operatorname{Rat}(\varGamma)$ denote the set of all rational functions
on $\varGamma$. For rational functions $f,g\in\operatorname{Rat}(\varGamma)$
and a point $x\in\varGamma\setminus\varGamma_{\infty}$, we define

$\displaystyle(f\oplus
g)(x):=\operatorname{max}\\{f(x),g(x)\\}\quad\text{and}\quad(f\odot
g)(x):=f(x)+g(x).$

We extend $f\oplus g$ and $f\odot g$ to points at infinity to be continuous on
the whole of $\varGamma$. Then both are rational functions on $\varGamma$.
Note that for any $f\in\operatorname{Rat}(\varGamma)$, we have

$\displaystyle f\oplus(-\infty)=(-\infty)\oplus f=f$

and

$\displaystyle f\odot(-\infty)=(-\infty)\odot f=-\infty.$

Then $\operatorname{Rat}(\varGamma)$ becomes a semifield with these two
operations. Also, $\operatorname{Rat}(\varGamma)$ becomes a
$\boldsymbol{T}$-algebra and a semifield over $\boldsymbol{T}$ with the
natural inclusion
$\boldsymbol{T}\hookrightarrow\operatorname{Rat}(\varGamma)$. Note that for
$f,g\in\operatorname{Rat}(\varGamma)$, $f=g$ means that $f(x)=g(x)$ for any
$x\in\varGamma$.

###  2.7. Morphisms between tropical curves

Let $\varphi:\varGamma\to\varGamma^{\prime}$ be a continuous map between
tropical curves. Then $\varphi$ is a morphism if there exist loopless models
$(G,l)$ and $(G^{\prime},l^{\prime})$ for $\varGamma$ and
$\varGamma^{\prime}$, respectively, such that $\varphi$ can be regarded as a
map $V(G)\cup E(G)\to V(G^{\prime})\cup E(G^{\prime})$ satisfying
$\varphi(V(G))\subset V(G^{\prime})$ and for $e\in\varphi(E(G))$, there exists
a nonnegative integer $\operatorname{deg}_{e}(\varphi)$ such that for any
points $x,y$ of $e$,
$\operatorname{dist}_{\varphi(e)}(\varphi(x),\varphi(y))=\operatorname{deg}_{e}(\varphi)\cdot\operatorname{dist}_{e}(x,y)$,
where $\operatorname{dist}_{\varphi(e)}(\varphi(x),\varphi(y))$ (resp.
$\operatorname{dist}_{e}(x,y)$) denotes the distance between $\varphi(x)$ and
$\varphi(y)$ in $\varphi(e)$ (resp. $x$ and $y$ in $e$). A map between
tropical curves $\varphi:\varGamma\to\varGamma^{\prime}$ is an isomorphism if
$\varphi$ is a bijective morphism and the inverse map $\varphi^{-1}$ of
$\varphi$ is also a morphism. By this definition, $\varphi$ is an isomorphism
if and only if $\varphi$ is continuous on the whole of $\varGamma$ and is a
local isometry on $\varGamma\setminus\varGamma_{\infty}$. Then $\varGamma$ is
isomorphic to $\varGamma^{\prime}$. For a morphism
$\varphi:\varGamma\to\varGamma^{\prime}$, the pull-back
$\varphi^{\ast}:\operatorname{Rat}(\varGamma^{\prime})\to\operatorname{Rat}(\varGamma);f^{\prime}\mapsto
f^{\prime}\circ\varphi$ is a $\boldsymbol{T}$-algebra homomorphism (cf. [14,
Proposition 3.1]).

###  2.8. Linear systems

Let $\varGamma$ be a tropical curve. An element of the free abelian group
$\operatorname{Div}(\varGamma)$ generated by all points of $\varGamma$ is
called a divisor on $\varGamma$. For a divisor $D$ on $\varGamma$, its degree
$\operatorname{deg}(D)$ is defined by the sum of the coefficients over all
points of $\varGamma$. We write the coefficient at $x$ as $D(x)$. A divisor
$D$ on $\varGamma$ is said to be effective if $D(x)\geq 0$ for any $x$ in
$\varGamma$. For any
$f\in\operatorname{Rat}(\varGamma)\setminus\\{-\infty\\}$, the sum
$\operatorname{ord}_{x}(f)$ is zero for all $x$ but a finite number of points
of $\varGamma$, and thus

$\operatorname{div}(f):=\sum_{x\in\varGamma}\operatorname{ord}_{x}(f)\cdot x$

is a divisor on $\varGamma$, which is called the principal divisor defined by
$f$. Remark that the constant function of $-\infty$ on $\varGamma$ does not
define a principal divisor. Two divisors $D$ and $E$ on $\varGamma$ are said
to be linearly equivalent if $D-E$ is a principal divisor. For a divisor $D$
on $\varGamma$, the complete linear system $|D|$ is defined by the set of
effective divisors on $\varGamma$ which are linearly equivalent to $D$. For a
divisor $D$ on $\varGamma$, let $R(D)$ be the set of rational functions
$f\in\operatorname{Rat}(\varGamma)\setminus\\{-\infty\\}$ such that
$D+\operatorname{div}(f)$ is effective together with $-\infty$. [10, Corollary
9] states that $R(D)$ is finitely generated as a $\boldsymbol{T}$-module and
we can choose up extremals from every finite generating set to be a minimal
generating set. Here an extremal of $R(D)$ is an element $f$ of $R(D)$ such
that $f=g_{1}\oplus g_{2}$ implies $f=g_{1}$ or $f=g_{2}$ for $g_{1},g_{2}\in
R(D)$. A characterization of extremals is given by [10, Lemma 5].

##  3\. Main results

In this section, we prove all assertions in Section 1. To do so, we first deal
with finitely generated congruences on
$\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$. For a point
$x\in\boldsymbol{R}^{n}$, let $(x)_{i}$ denote the $i$-th component of $x$.

######  Lemma 3.1.

For $(f,g)\in\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}^{2}$, the
congruence variety $\boldsymbol{V}((f,g))$ is a finite union of
$\boldsymbol{R}$-rational polyhedral sets.

###### Proof.

If $\boldsymbol{V}((f,g))$ is empty or $\boldsymbol{R}^{n}$, then it is clear.

We assume that $\boldsymbol{V}((f,g))$ is not either empty or
$\boldsymbol{R}^{n}$. By assumption, $f,g\not=-\infty$ and $f\not=g$. Since
$f,g\in\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}\setminus\\{-\infty\\}$,
there exist
$f_{1},f_{2},g_{1},g_{2}\in\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}\setminus\\{-\infty\\}$
such that $f=f_{1}\odot f_{2}^{\odot(-1)}$ and $g=g_{1}\odot
g_{2}^{\odot(-1)}$. It is easy to check that for any $x\in\boldsymbol{R}^{n}$,
$x\in\boldsymbol{V}((f,g))$ if and only if $x\in\boldsymbol{V}((f_{1}\odot
g_{2},g_{1}\odot f_{2}))$. Hence we can assume that
$f,g\in\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}\setminus\\{-\infty\\}$.
Let $f^{\prime}$ and $g^{\prime}$ be representatives of $f$ and $g$ in
$\boldsymbol{T}[X_{1},\ldots,X_{n}]\setminus\\{-\infty\\}$ resprectively. By
definition, we have
$\boldsymbol{V}((f,g))=\boldsymbol{V}((f^{\prime},g^{\prime}))_{0}\cap\boldsymbol{R}^{n}$.

Let $f^{\prime}=\bigoplus_{\boldsymbol{i}=(i_{1},\ldots,i_{n})\in
A}a_{\boldsymbol{i}}\odot X_{1}^{\odot i_{1}}\odot\cdots\odot X_{n}^{\odot
i_{n}}$ and $g^{\prime}=\bigoplus_{\boldsymbol{j}=(j_{1},\ldots,j_{n})\in
B}b_{\boldsymbol{j}}\odot X_{1}^{\odot j_{1}}\odot\cdots\odot X_{n}^{\odot
j_{n}}$ for some $a_{\boldsymbol{i}},b_{\boldsymbol{j}}\in\boldsymbol{R}$ and
some finite subsets $A,B\subset\left(\boldsymbol{Z}_{\geq 0}\right)^{n}$,
where $\boldsymbol{i}\in A$ and $\boldsymbol{j}\in B$. For any
$x\in\boldsymbol{R}^{n}$, we have

$\displaystyle x\in\boldsymbol{V}((f^{\prime},g^{\prime}))_{0}$
$\displaystyle\iff$ $\displaystyle f^{\prime}(x)=g^{\prime}(x)$
$\displaystyle\iff$ $\displaystyle\text{there exist
}\boldsymbol{i}=(i_{1},\ldots,i_{n})\in
A,\boldsymbol{j}=(j_{1},\ldots,j_{n})\in B\text{ such that}$
$\displaystyle\text{for any }\boldsymbol{k}=(k_{1},\ldots,k_{n})\in
A,\boldsymbol{l}=(l_{1},\ldots,l_{n})\in B,$ $\displaystyle\hskip
15.0pta_{\boldsymbol{i}}+i_{1}(x)_{1}+\cdots+i_{n}(x)_{n}=b_{\boldsymbol{j}}+j_{1}(x)_{1}+\cdots+j_{n}(x)_{n}$
$\displaystyle\geq
a_{\boldsymbol{k}}+k_{1}(x)_{1}+\cdots+k_{n}(x)_{n},b_{\boldsymbol{l}}+l_{1}(x)_{1}+\cdots+l_{n}(x)_{n}$
$\displaystyle\iff$ $\displaystyle\text{there exist
}\boldsymbol{i}=(i_{1},\ldots,i_{n})\in
A,\boldsymbol{j}=(j_{1},\ldots,j_{n})\in B\text{ such that}$
$\displaystyle\text{for any }\boldsymbol{k}=(k_{1},\ldots,k_{n})\in
A,\boldsymbol{l}=(l_{1},\ldots,l_{n})\in B,$
$\displaystyle(a_{\boldsymbol{i}}-b_{\boldsymbol{j}})+(i_{1}-j_{1})(x)_{1}+\cdots+(i_{n}-j_{n})(x)_{n}=0,$
$\displaystyle(a_{\boldsymbol{i}}-a_{\boldsymbol{k}})+(i_{1}-k_{1})(x)_{1}+\cdots+(i_{n}-k_{n})(x)_{n}\geq
0,$
$\displaystyle(b_{\boldsymbol{j}}-b_{\boldsymbol{l}})+(j_{1}-l_{1})(x)_{1}+\cdots+(j_{n}-l_{n})(x)_{n}\geq
0.$

Therefore we have

$\displaystyle\boldsymbol{V}((f^{\prime},g^{\prime}))_{0}$
$\displaystyle\cap\boldsymbol{R}^{n}$ $\displaystyle=$
$\displaystyle\bigcup_{\begin{subarray}{c}\boldsymbol{i}=(i_{1},\ldots,i_{n})\in
A\\\ \boldsymbol{j}=(j_{1},\ldots,j_{n})\in B\end{subarray}}\Bigg{[}$
$\displaystyle\\{x\in\boldsymbol{R}^{n}\,|\,(a_{\boldsymbol{i}}-b_{\boldsymbol{j}})+(i_{1}-j_{1})(x)_{1}+\cdots+(i_{n}-j_{n})(x)_{n}\geq
0\\}$
$\displaystyle\cap\\{x\in\boldsymbol{R}^{n}\,|\,(a_{\boldsymbol{i}}-b_{\boldsymbol{j}})+(i_{1}-j_{1})(x)_{1}+\cdots+(i_{n}-j_{n})(x)_{n}\leq
0\\}$ $\displaystyle\cap\bigcap_{\boldsymbol{k}=(k_{1},\ldots,k_{n})\in
A}\\{x\in\boldsymbol{R}^{n}\,|\,(a_{\boldsymbol{i}}-a_{\boldsymbol{k}})+(i_{1}-k_{1})(x)_{1}+\cdots+(i_{n}-k_{n})(x)_{n}\geq
0\\}$ $\displaystyle\cap\bigcap_{\boldsymbol{l}=(l_{1},\ldots,l_{n})\in
B}\\{x\in\boldsymbol{R}^{n}\,|\,(b_{\boldsymbol{j}}-b_{\boldsymbol{l}})+(j_{1}-l_{1})(x)_{1}+\cdots+(j_{n}-l_{n})(x)_{n}\geq
0\\}\Bigg{]}.$

This completes the proof. ∎

######  Lemma 3.2.

Let $E$ be a congruence on $\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$.
For
$f,g\in\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}\setminus\\{-\infty\\}$,
$(f,g)\in E$ if and only if $\left(f\odot g^{\odot(-1)}\oplus
0,0\right),\left(f^{\odot(-1)}\odot g\oplus 0,0\right)\in E$.

###### Proof.

First we note that since $f,g\not=-\infty$, the inverses
$f^{\odot(-1)},g^{\odot(-1)}$ exist.

We assume that $(f,g)\in E$ holds. As $E$ is a congruence, we have

$\displaystyle\left(f\odot g^{\odot(-1)}\oplus 0,0\right)=\left(f\odot
g^{\odot(-1)}\oplus 0,g\odot g^{\odot(-1)}\oplus 0\right)\in E,$
$\displaystyle\left(f^{\odot(-1)}\odot g\oplus
0,0\right)=\left(f^{\odot(-1)}\odot g\oplus 0,f^{\odot(-1)}\odot f\oplus
0\right)\in E.$

Conversely we assume that $\left(f\odot g^{\odot(-1)}\oplus
0,0\right),\left(f^{\odot(-1)}\odot g\oplus 0,0\right)\in E$ hold. Because

$\displaystyle(f\oplus g,g)$ $\displaystyle=\left(\left(f\odot
g^{\odot(-1)}\oplus 0\right)\odot g,0\odot g\right)\in E,$
$\displaystyle(f\oplus g,f)$ $\displaystyle=\left(\left(f^{\odot(-1)}\odot
g\oplus 0\right)\odot f,0\odot f\right)\in E,$

we have $(f,g)\in E$ by symmetry and transitivity. ∎

######  Lemma 3.3 (cf. [15, Lemma 3.4]).

For any semiring $S$, the semiring $S^{2}$ is generated by $(1,0)$.

###### Proof.

For any $x\in S$, we have $(x,0)=(x\cdot 1,x\cdot 0)\in\langle(1,0)\rangle$.
Thus, by symmetry and transitivity, for any $x,y\in S$,
$(x,y)\in\langle(1,0)\rangle$ holds, which means that
$S^{2}=\langle(1,0)\rangle$. ∎

######  Proposition 3.4 (cf. [18, Proposition 3.10]).

For a congruence $E$ on $\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$, $E$
is finitely generated as a congruence if and only if there exists
$f\in\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$ such that
$E=\langle(f,0)\rangle$. In particular, if $E$ is proper, then $f$ above can
be taken as $f\geq 0$.

###### Proof.

The if part is clear. We shall show the only if part. If $E$ is trivial, the
assertion is clear, and if $E$ is improper, we have the conclusion by Lemma
3.3. Assume that $E$ is nontrivial and proper. Assume that $E$ is finitely
generated as a congruence. Then there exist
$(f_{1},g_{1}),\ldots,(f_{m},g_{m})\in\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}^{2}$
that generate $E$, i.e., $\langle(f_{1},g_{1}),\ldots,(f_{m},g_{m})\rangle=E$.
If there exists $j$ such that $f_{j}\not=-\infty$ and $g_{j}=-\infty$, then,
since the inverse $f_{j}^{\odot(-1)}$ exists, we have
$(0,-\infty)=\left(f_{j}\odot f_{j}^{\odot(-1)},g_{j}\odot
f_{j}^{\odot(-1)}\right)\in E$, which is a contradiction by Lemma 3.3. Hence,
without loss of generality, we can assume that for any $i$,
$(f_{i},g_{i})\not\in\Delta$ and $f_{i},g_{i}\not=-\infty$. By Lemma 3.2,

$\displaystyle\left(f_{1}\odot g_{1}^{\odot(-1)}\oplus
0,0\right),\left(f_{1}^{\odot(-1)}\odot g_{1}\oplus 0,0\right),$
$\displaystyle\ldots,\left(f_{m}\odot g_{m}^{\odot(-1)}\oplus
0,0\right),\left(f_{m}^{\odot(-1)}\odot g_{m}\oplus 0,0\right)$

are also generators of $E$. Let $h_{i}:=f_{i}\odot g_{i}^{\odot(-1)}\oplus 0$
and $h^{\prime}_{i}:=f_{i}^{\odot(-1)}\odot g_{i}\oplus 0$ for each $i$. Then
$h_{i}\geq 0$ and $h^{\prime}_{i}\geq 0$, and hence we have

$\displaystyle h_{1}\odot\cdots\odot h_{m}\odot h_{1}^{\prime}\odot\cdots\odot
h_{m}^{\prime}\geq h_{1}\oplus\cdots\oplus h_{m}\oplus
h_{1}^{\prime}\oplus\cdots\oplus h_{m}^{\prime}\geq h_{i},h_{i}^{\prime}\geq
0$

for any $i$. By [17, Proposition 2.2(iii)],

$\displaystyle(h_{i},0),(h_{i}^{\prime},0)$
$\displaystyle\in\langle(h_{1}\oplus\cdots\oplus h_{m}\oplus
h_{1}^{\prime}\oplus\cdots\oplus h_{m}^{\prime},0)\rangle$
$\displaystyle\subset\langle(h_{1}\odot\cdots\odot h_{m}\odot
h_{1}^{\prime}\odot\cdots\odot h_{m}^{\prime},0)\rangle$

hold for any $i$. Since
$(h_{1},0),\ldots,(h_{m},0),(h_{1}^{\prime},0),\ldots,(h_{m}^{\prime},0)$ are
generators of $E$, the converse inclusions are clear. Hence

$\displaystyle E$ $\displaystyle=\langle(h_{1}\odot\cdots\odot h_{m}\odot
h_{1}^{\prime}\odot\cdots\odot h_{m}^{\prime},0)\rangle$
$\displaystyle=\langle(h_{1}\oplus\cdots\oplus h_{m}\oplus
h_{1}^{\prime}\oplus\cdots\oplus h_{m}^{\prime},0)\rangle$

hold. ∎

By Lemma 3.1 and Proposition 3.4, we have the following corollary:

######  Corollary 3.5.

For a finitely generated congruence $E$ on
$\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$, the congruence variety
$\boldsymbol{V}(E)$ is a finite union of $\boldsymbol{R}$-rational polyhedral
sets.

Next, we prove that the congruence on
$\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$ defined by a finite union of
$\boldsymbol{R}$-rational polyhedral sets is finitely generated as a
congruence.

######  Lemma 3.6.

Let $V$ be a nonempty subset of $\boldsymbol{R}^{n}$. For any
$(g,0)\in\boldsymbol{E}\left(\overline{V}\right)$ such that $g\geq 0$, there
exists $N\in\boldsymbol{Z}_{>0}$ such that
$\operatorname{dist}\left(x,\overline{V}\right)^{\odot N}\geq g(x)$ holds for
any $x\in\boldsymbol{R}^{n}$.

###### Proof.

As $(g,0)\in\boldsymbol{E}\left(\overline{V}\right)$, for any
$x\in\overline{V}$, we have $g(x)=0$. Let
$x\in\boldsymbol{R}^{n}\setminus\overline{V}$ and $y$ be a point in
$\overline{V}$ satisfying
$\operatorname{dist}(x,y)=\operatorname{dist}\left(x,\overline{V}\right)$. Let
$z:=x-y$. Then we have
$\operatorname{dist}\left(x,\overline{V}\right)=\sqrt{(z)_{1}^{2}+\cdots+(z)_{n}^{2}}\geq|(z)_{1}|,\ldots,|(z)_{n}|$.
Hence $\operatorname{dist}\left(x,\overline{V}\right)^{\odot
n}\geq|(z)_{1}|\odot\cdots\odot|(z)_{n}|$ holds. Because $\overline{V}$ is
nonempty and $(g,0)\in\boldsymbol{E}\left(\overline{V}\right)$, $g$ is not
$-\infty$. Thus there exist
$g_{1},g_{2}\in\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}\setminus\\{-\infty\\}$
such that $g=g_{1}\odot g_{2}^{\odot(-1)}$. Let
$g_{i}^{\prime}=\bigoplus_{\boldsymbol{j}_{i}=(j_{i,1},\ldots,j_{i,n})\in
B_{i}}b_{\boldsymbol{j}_{i}}\odot X_{1}^{\odot j_{i,1}}\odot\cdots\odot
X_{n}^{\odot
j_{i,n}}\in\boldsymbol{T}[X_{1},\ldots,X_{n}]\setminus\\{-\infty\\}$ be a
representative of $g_{i}$, where $B_{i}$ is a finite subset of
$\left(\boldsymbol{Z}_{\geq 0}\right)^{n}$ and
$b_{\boldsymbol{j}_{i}}\in\boldsymbol{R}$ for
$\boldsymbol{j}_{i}=(j_{i,1},\ldots,j_{i,n})\in B_{i}$. Let
$g_{1}(x)=b_{\boldsymbol{j}_{1}}\odot(x)_{1}^{\odot
j_{1,1}}\odot\cdots\odot(x)_{n}^{\odot j_{1,n}}$ for some
$\boldsymbol{j}_{1}=(j_{1,1},\ldots,j_{1,n})\in B_{1}$ and
$g_{2}(y)=b_{\boldsymbol{j}_{2}}\odot(y)_{1}^{\odot
j_{2,1}}\odot\cdots\odot(y)_{n}^{\odot j_{2,n}}$ for some
$\boldsymbol{j}_{2}=(j_{2,1},\ldots,j_{2,n})\in B_{2}$. We have

$\displaystyle g_{1}(x)=g_{1}(y+z)$
$\displaystyle=b_{\boldsymbol{j}_{1}}\odot(y+z)_{1}^{\odot
j_{1,1}}\odot\cdots\odot(y+z)_{n}^{\odot j_{1,n}}$
$\displaystyle=b_{\boldsymbol{j}_{1}}\odot(y)_{1}^{\odot
j_{1,1}}\odot\cdots\odot(y)_{n}^{\odot j_{1,n}}\odot(z)_{1}^{\odot
j_{1,1}}\odot\cdots\odot(z)_{n}^{\odot j_{1,n}}$ $\displaystyle\leq
g_{1}(y)\odot(z)_{1}^{\odot j_{1,1}}\odot\cdots\odot(z)_{n}^{\odot j_{1,n}}$

and

$\displaystyle g_{2}(y)$
$\displaystyle=b_{\boldsymbol{j}_{2}}\odot(y)_{1}^{\odot
j_{2,1}}\odot\cdots\odot(y)_{n}^{\odot j_{2,n}}$
$\displaystyle=b_{\boldsymbol{j}_{2}}\odot(y+z-z)_{1}^{\odot
j_{2,1}}\odot\cdots\odot(y+z-z)_{n}^{\odot j_{2,n}}$
$\displaystyle=b_{\boldsymbol{j}_{2}}\odot(y+z)_{1}^{\odot
j_{2,1}}\odot\cdots\odot(y+z)_{n}^{\odot j_{2,n}}\odot(-z)_{1}^{\odot
j_{2,1}}\odot\cdots\odot(-z)_{n}^{\odot j_{2,n}}$ $\displaystyle\leq
g_{2}(y+z)\odot(-z)_{1}^{\odot j_{2,1}}\odot\cdots\odot(-z)_{n}^{\odot
j_{2,n}}.$

Since $0=g(y)=g_{1}(y)\odot g_{2}(y)^{\odot(-1)}$, we have
$g_{1}(y)=g_{2}(y)\in\boldsymbol{R}$. Therefore

$\displaystyle g(x)=g(y+z)$ $\displaystyle=g_{1}(y+z)\odot
g_{2}(y+z)^{\odot(-1)}$ $\displaystyle\leq g_{1}(y)\odot(z)_{1}^{\odot
j_{1,1}}\odot\cdots\odot(z)_{n}^{\odot j_{1,n}}\odot g_{2}(y+z)^{\odot(-1)}$
$\displaystyle=g_{2}(y)\odot(z)_{1}^{\odot
j_{1,1}}\odot\cdots\odot(z)_{n}^{\odot j_{1,n}}\odot g_{2}(y+z)^{\odot(-1)}$
$\displaystyle\leq(z)_{1}^{\odot j_{1,1}}\odot\cdots\odot(z)_{n}^{\odot
j_{1,n}}\odot(-z)_{1}^{\odot j_{2,1}}\odot\cdots\odot(-z)_{n}^{\odot j_{2,n}}$
$\displaystyle=(z)_{1}^{\odot(j_{1,1}-j_{2,1})}\odot\cdots\odot(z)_{n}^{\odot(j_{1,n}-j_{2,n})}$
$\displaystyle\leq\operatorname{max}\left\\{(z)_{1}^{\odot(j_{1,1}^{\prime}-j_{2,1}^{\prime})}\odot\cdots\odot(z)_{n}^{\odot(j_{1,n}^{\prime}-j_{2,n}^{\prime})}\,\middle|\,\begin{aligned}
(j_{1,1}^{\prime},\ldots,j_{1,n}^{\prime})\in B_{1}\\\
(j_{2,1}^{\prime},\ldots,j_{2,n}^{\prime})\in B_{2}\end{aligned}\right\\}$

hold. The inequality $\operatorname{dist}\left(x,\overline{V}\right)^{\odot
nk}\geq g(x)$ holds for $k\in\boldsymbol{Z}_{>0}$ greater than
$\operatorname{max}\left\\{|j_{1,1}^{\prime}-j_{2,1}^{\prime}|,\ldots,|j_{1,n}^{\prime}-j_{2,n}^{\prime}|\,|\,(j_{1,1}^{\prime},\ldots,j_{1,n}^{\prime})\in
B_{1},(j_{2,1}^{\prime},\ldots,j_{2,n}^{\prime})\in B_{2}\right\\}$. We note
that this $k$ is independent of the choice of $x$. In conclusion, for $N=nk$,
we have $\operatorname{dist}\left(x,\overline{V}\right)^{\odot N}\geq g(x)$
for any $x\in\boldsymbol{R}^{n}$. ∎

######  Corollary 3.7.

Let $V$ be a nonempty subset of $\boldsymbol{R}^{n}$. For
$f\in\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$, assume that $(1)$
$\boldsymbol{V}((f,0))=\overline{V}$ holds and $(2)$ there exists
$k^{\prime}\in\boldsymbol{Z}_{>0}$ such that for any $x\in\boldsymbol{R}^{n}$,
$f(x)^{\odot k^{\prime}}\geq\operatorname{dist}\left(x,\overline{V}\right)$
holds. Then for any $(g,0)\in\boldsymbol{E}\left(\overline{V}\right)$ such
that $g\geq 0$, there exists $N^{\prime}\in\boldsymbol{Z}_{>0}$ such that
$f^{\odot N^{\prime}}\geq g$ and
$\boldsymbol{E}\left(\overline{V}\right)=\langle(f,0)\rangle$ holds.

###### Proof.

By $(1)$, we have
$\boldsymbol{E}\left(\overline{V}\right)\supset\langle(f,0)\rangle$. For $N$
in Lemma 3.6, by $(2)$, we have $0\leq
g(x)\leq\operatorname{dist}\left(x,\overline{V}\right)^{\odot N}\leq
f(x)^{\odot Nk^{\prime}}$ for any $x\in\boldsymbol{R}^{n}$, thus $0\leq g\leq
f^{\odot N^{\prime}}$ for $N^{\prime}=Nk^{\prime}$. By [17, Proposition
2.2(iii)], this means that $(g,0)\in\langle(f,0)\rangle$. Since $V$ is
nonempty, by [15, Lemmas 3.4 and 3.7], if
$(h,-\infty)\in\boldsymbol{E}\left(\overline{V}\right)$, then $h=-\infty$, and
thus we have
$\boldsymbol{E}\left(\overline{V}\right)\subset\langle(f,0)\rangle$ by Lemma
3.2. ∎

######  Corollary 3.8.

Let $f\in\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$ be represented by
$a\odot X_{1}^{\odot i_{1}}\odot\cdots\odot X_{n}^{\odot i_{n}}\oplus
0\in\boldsymbol{T}\left[X_{1}^{\pm 1},\ldots,X_{n}^{\pm 1}\right]$. If
$a\in\boldsymbol{R}$ and at least one of $i_{1},\ldots,i_{n}$ is not zero,
then $f(x)\geq\operatorname{dist}(x,\boldsymbol{V}((f,0)))$ for any
$x\in\boldsymbol{R}^{n}$ and
$\boldsymbol{E}(\boldsymbol{V}((f,0)))=\langle(f,0)\rangle$ hold.

###### Proof.

By assumption, $\sqrt{i_{1}^{2}+\cdots+i_{n}^{2}}\geq 1$ holds. By the
definition of $f$, $\boldsymbol{V}((f,0))$ is nonempty, and for any
$x\in\boldsymbol{R}^{n}$, $f(x)$ is
$\operatorname{dist}(x,\boldsymbol{V}((f,0)))$ multiplied (in the usual
meaning) by the positive real number $\sqrt{i_{1}^{2}+\cdots+i_{n}^{2}}$.
Hence $f(x)\geq\operatorname{dist}(x,\boldsymbol{V}((f,0)))$ holds. By
Corollary 3.7, we have
$\boldsymbol{E}(\boldsymbol{V}((f,0)))=\langle(f,0)\rangle$. ∎

######  Lemma 3.9.

Let $m\in\boldsymbol{Z}_{\geq 1}$. For $j=1,\ldots,m$, let
$f_{j}\in\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$ be represented by
$a_{j}\odot X_{1}^{\odot i_{j,1}}\odot\cdots\odot X_{n}^{\odot i_{j,n}}\oplus
0\in\boldsymbol{T}\left[X_{1}^{\pm 1},\ldots,X_{n}^{\pm 1}\right]$. If
$a_{j}\in\boldsymbol{R}$ and at least one of $i_{j,1},\ldots,i_{j,n}$ is not
zero for any $j$ and $\bigcap_{j=1}^{m}\boldsymbol{V}((f_{j},0))$ is nonempty,
then the following hold:

$(1)$ $\boldsymbol{V}((f_{1}\odot\cdots\odot
f_{m},0))=\boldsymbol{V}((f_{1}\oplus\cdots\oplus
f_{m},0))=\bigcap_{j=1}^{m}\boldsymbol{V}((f_{j},0))$,

$(2)$ there exists $k\in\boldsymbol{Z}_{>0}$ such that
$\left(f_{1}(x)\odot\cdots\odot f_{m}(x)\right)^{\odot
k}\geq\operatorname{dist}\left(x,\bigcap_{j=1}^{m}\boldsymbol{V}((f_{j},0))\right)$,
$\left(f_{1}(x)\oplus\cdots\oplus f_{m}(x)\right)^{\odot
k}\geq\operatorname{dist}\left(x,\bigcap_{j=1}^{m}\boldsymbol{V}((f_{j},0))\right)$
hold for any $x\in\boldsymbol{R}^{n}$, and

$(3)$
$\boldsymbol{E}\left(\bigcap_{j=1}^{m}\boldsymbol{V}((f_{j},0))\right)=\langle(f_{1}\odot\cdots\odot
f_{m},0)\rangle=\langle(f_{1}\oplus\cdots\oplus f_{m},0)\rangle$.

###### Proof.

$(1)$ and $(2)$ imply $(3)$ by Corollary 3.7.

Since $f_{j}\geq 0$, $f_{1}\odot\cdots\odot f_{m}\geq f_{1}\oplus\cdots\oplus
f_{m}\geq 0$ hold, and thus $\langle(f_{1}\odot\cdots\odot
f_{m},0)\rangle\supset\langle(f_{1}\oplus\cdots\oplus f_{m},0)\rangle$ holds
by [17, Proposition 2.2(iii)].

Conversely, as $f_{1}\oplus\cdots\oplus f_{m}\geq f_{j}$, we have
$(f_{1}\oplus\cdots\oplus f_{m})^{\odot m}\geq f_{1}\odot\cdots\odot f_{m}$,
and hence $\langle(f_{1}\odot\cdots\odot
f_{m},0)\rangle\subset\langle(f_{1}\oplus\cdots\oplus f_{m},0)\rangle$ by [17,
Proposition 2.2(iii)].

Because for any $x\in\bigcap_{j=1}^{m}\boldsymbol{V}((f_{j},0))$,
$f_{1}(x)=\cdots=f_{m}(x)=0$ hold, we have
$\boldsymbol{V}((f_{1}\odot\cdots\odot
f_{m},0))\supset\bigcap_{j=1}^{m}\boldsymbol{V}((f_{j},0))$.

Conversely, for any $x\in\boldsymbol{V}((f_{1}\odot\cdots\odot f_{m},0))$,
since $f_{i}\geq 0$ and $f_{1}(x)\odot\cdots\odot f_{m}(x)=0$, we have
$f_{1}(x)=\cdots=f_{m}(x)=0$ hold. Thus $\boldsymbol{V}((f_{1}\odot\cdots\odot
f_{m},0))\subset\bigcap_{j=1}^{m}\boldsymbol{V}((f_{j},0))$.

$(2)$ follows from Corollary 2.3 and Lemma 2.4. ∎

######  Lemma 3.10.

Let $V$ and $W$ be nonempty subsets of $\boldsymbol{R}^{n}$. Assume
$f,g\in\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$ satisfy the following
conditions: $\boldsymbol{V}((f,0))=\overline{V}$,
$\boldsymbol{V}((g,0))=\overline{W}$, and there exist
$k_{1},k_{2}\in\boldsymbol{Z}_{>0}$ such that $f(x)^{\odot
k_{1}}\geq\operatorname{dist}\left(x,\overline{V}\right)$ and $g(x)^{\odot
k_{2}}\geq\operatorname{dist}\left(x,\overline{W}\right)$ hold for any
$x\in\boldsymbol{R}^{n}$. Then the following hold:

$(1)$ $\boldsymbol{V}\left(\left(\left(f^{\odot(-1)}\oplus
g^{\odot(-1)}\right)^{\odot(-1)},0\right)\right)=\overline{V}\cup\overline{W}$,

$(2)$ for $k:=\operatorname{max}\\{k_{1},k_{2}\\}$,
$\left(f(x)^{\odot(-1)}\oplus
g(x)^{\odot(-1)}\right)^{\odot(-k)}\geq\operatorname{dist}\left(x,\overline{V}\cup\overline{W}\right)$
holds for any $x\in\boldsymbol{R}^{n}$, and

$(3)$
$\boldsymbol{E}\left(\overline{V}\cup\overline{W}\right)=\left\langle\left(\left(f^{\odot(-1)}\oplus
g^{\odot(-1)}\right)^{\odot(-1)},0\right)\right\rangle$.

###### Proof.

$(1)$ and $(2)$ imply $(3)$ by Corollary 3.7.

For any $x\in\overline{V}$ and $y\in\overline{W}$, we have $f(x)=0$ and
$g(y)=0$. Thus for any $z\in\overline{V}\cup\overline{W}$, at least one of
$f(z)$ and $g(z)$ is zero. By assumption, $f\geq 0$ and $g\geq 0$ hold, and
hence we have $f^{\odot(-1)}\leq 0$ and $g^{\odot(-1)}\leq 0$. From this,
$f(z)^{\odot(-1)}\oplus g(z)^{\odot(-1)}=0$ holds. Thus we have
$\boldsymbol{V}\left(\left(\left(f^{\odot(-1)}\oplus
g^{\odot(-1)}\right)^{\odot(-1)},0\right)\right)\supset\overline{V}\cup\overline{W}$
and we also have $\left(\left(f^{\odot(-1)}\oplus
g^{\odot(-1)}\right)^{\odot(-1)},0\right)\in\boldsymbol{E}\left(\overline{V}\cup\overline{W}\right)$,
and thus $\left\langle\left(\left(f^{\odot(-1)}\oplus
g^{\odot(-1)}\right)^{\odot(-1)},0\right)\right\rangle\subset\boldsymbol{E}\left(\overline{V}\cup\overline{W}\right)$.
Since $\left(f(x)^{\odot(-1)}\oplus g(x)^{\odot(-1)}\right)^{\odot(-1)}=0$
implies $f(x)=0$ or $g(x)=0$, we have
$\boldsymbol{V}\left(\left(\left(f^{\odot(-1)}\oplus
g^{\odot(-1)}\right)^{\odot(-1)},0\right)\right)\subset\overline{V}\cup\overline{W}$.

Because $\overline{V}\cup\overline{W}\supset\overline{V},\overline{W}$, we
have
$\operatorname{dist}\left(x,\overline{V}\cup\overline{W}\right)\leq\operatorname{dist}\left(x,\overline{V}\right),\operatorname{dist}\left(x,\overline{W}\right)$.
Hence for any $x\in\boldsymbol{R}^{n}$, if $f(x)\geq g(x)$, then

$\displaystyle\left(f(x)^{\odot(-1)}\oplus
g(x)^{\odot(-1)}\right)^{\odot(-k)}=g(x)^{\odot
k}\geq\operatorname{dist}\left(x,\overline{W}\right)\geq\operatorname{dist}\left(x,\overline{V}\cup\overline{W}\right)$

hold; if $f(x)\leq g(x)$, then

$\displaystyle\left(f(x)^{\odot(-1)}\oplus
g(x)^{\odot(-1)}\right)^{\odot(-k)}=f(x)^{\odot
k}\geq\operatorname{dist}\left(x,\overline{V}\right)\geq\operatorname{dist}\left(x,\overline{V}\cup\overline{W}\right)$

hold. ∎

Since
$\boldsymbol{E}(\varnothing)=\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}^{2}$,
by Corollary 3.5 and Lemmas 3.3, 3.9, 3.10, we have Theorem 1.1. Also
Corollary 1.2 follows from Corollary 3.5 and Theorem 1.1.

######  Lemma 3.11.

For
$f_{1},\ldots,f_{n}\in\overline{\boldsymbol{T}(Y_{1},\ldots,Y_{m})}\setminus\\{-\infty\\}$,
let
$\theta:\boldsymbol{R}^{m}\to\boldsymbol{R}^{n};y\mapsto(f_{1}(y),\ldots,f_{n}(y))$.
Then $\theta$ maps a $\boldsymbol{R}$-rational polyhedral set in
$\boldsymbol{R}^{m}$ to a finite union of $\boldsymbol{R}$-rational polyhedral
sets in $\boldsymbol{R}^{n}$.

###### Proof.

Each $f_{i}$ breaks $\boldsymbol{R}^{m}$ into finitely many closed subsets on
each which $f_{i}$ is integral affine. Since the number of $f_{i}$ is finite,
we can break $\boldsymbol{R}^{m}$ into finitely many closed subsets on each of
which each $f_{i}$ is integral affine. As such each closed sunset is also a
$\boldsymbol{R}$-rational polyhedral set, its image by $\theta$ is also a
$\boldsymbol{R}$-rational polyhedral set, and hence we have the conclusion. ∎

######  Corollary 3.12.

Let $V$ (resp. $W$) be a subset of $\boldsymbol{R}^{n}$ (resp.
$\boldsymbol{R}^{m}$). Let
$\psi:\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}/\boldsymbol{E}(V)\to\overline{\boldsymbol{T}(Y_{1},\ldots,Y_{m})}/\boldsymbol{E}(W)$
be a $\boldsymbol{T}$-algebra homomorphism and
$\pi_{\boldsymbol{E}(V)}:\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}\twoheadrightarrow\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}/\boldsymbol{E}(V)$
and
$\pi_{\boldsymbol{E}(W)}:\overline{\boldsymbol{T}(Y_{1},\ldots,Y_{m})}\twoheadrightarrow\overline{\boldsymbol{T}(Y_{1},\ldots,Y_{m})}/\boldsymbol{E}(W)$
the natural surjective $\boldsymbol{T}$-algebra homomorphisms. Fix an element
$f_{i}\in\pi_{\boldsymbol{E}(W)}^{-1}\left(\psi\left(\pi_{\boldsymbol{E}(V)}(X_{i})\right)\right)\setminus\\{-\infty\\}$
for any $i$. Let
$\theta:\boldsymbol{R}^{m}\to\boldsymbol{R}^{n};y\mapsto(f_{1}(y),\ldots,g_{n}(y))$.
If $\psi$ is injective, $\theta(\overline{W})$ is closed and
$\boldsymbol{E}(W)$ is finitely generated as a congruence, then
$\boldsymbol{E}(V)$ is finitely generated as a congruence. In particular, when
$\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}/\boldsymbol{E}(V)$ is
isomorphic to
$\overline{\boldsymbol{T}(Y_{1},\ldots,Y_{m})}/\boldsymbol{E}(W)$ as a
$\boldsymbol{T}$-algebra, $\boldsymbol{E}(V)$ is finitely generated as a
congruence if and only if so is $\boldsymbol{E}(W)$.

###### Proof.

There exist finitely many closed half-spaces
$H_{11},\ldots,H_{1k_{1}},H_{21},\ldots,H_{jk_{j}}$ in $\boldsymbol{R}^{n}$
that have integral normal vectors, such that
$\bigcup_{a=1}^{j}\bigcap_{b=1}^{k_{a}}H_{ab}=\overline{W}$ by Corollary 3.5.
By [15, Theorem 3.15], we have

$\displaystyle\overline{V}=\theta\left(\overline{W}\right)=\theta\left(\bigcup_{a=1}^{j}\bigcap_{b=1}^{k_{a}}H_{ab}\right)=\bigcup_{a=1}^{j}\theta\left(\bigcap_{b=1}^{k_{a}}H_{ab}\right).$

By Lemma 3.11 and Theorem 1.1, we have the conclusion. ∎

###### Proof of Corollary 1.4.

If $V$ is empty, then we have

$\displaystyle\langle(0,-\infty)\rangle_{\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}}=\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}^{2}=\boldsymbol{E}(\varnothing)_{1}$

by Lemma 3.3. For any
$(g,h)\in\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}^{2}$, we have
$(-\infty,-\infty)=(g\odot(-\infty),h\odot(-\infty))\in\Delta\subset\langle(0,-\infty)\rangle_{\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}}$,
and hence

$\displaystyle\boldsymbol{E}(\varnothing)_{1}=$
$\displaystyle~{}\left\\{(g,h)\in\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}^{2}\,\middle|\,\right.$
$\displaystyle~{}\left.\exists
k\in\boldsymbol{Z}_{>0}:\left(g\odot(-\infty)^{\odot k},h\odot(-\infty)^{\odot
k}\right)\in\langle(0,-\infty)\rangle_{\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}}\right\\}.$

Cleary $0\geq-\infty$ on $\boldsymbol{T}^{n}$.

Assume that $V$ is nonempty. Then, by Lemmas 3.2, 3.6, 3.9 and 3.10, there
exists
$f\in\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}\setminus\\{-\infty\\}$ such
that $\boldsymbol{E}(V)=\langle(f,0)\rangle$, $f\geq 0$ on
$\boldsymbol{R}^{n}$ and for any
$(g,h)\in\boldsymbol{E}(V)\setminus\\{(-\infty,-\infty)\\}$, there exists
$k\in\boldsymbol{Z}_{>0}$ such that

$\displaystyle 0\leq g\odot h^{\odot(-1)}\oplus 0\leq f^{\odot k}$

and

$\displaystyle 0\leq g^{\odot(-1)}\odot h\oplus 0\leq f^{\odot k}$

on $\boldsymbol{R}^{n}$. There exist
$f_{1},f_{2}\in\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}\setminus\\{-\infty\\}$
such that $f=f_{1}\odot f_{2}^{\odot(-1)}$. Therefore we have

$\displaystyle h\odot f_{2}^{\odot k}\leq(g\oplus h)\odot f_{2}^{\odot k}\leq
h\odot f_{1}^{\odot k}$

and

$\displaystyle g\odot f_{2}^{\odot k}\leq(g\oplus h)\odot f_{2}^{\odot k}\leq
g\odot f_{1}^{\odot k}$

on $\boldsymbol{R}^{n}$. Note that, by [7, Lemma 3.7.4], these inequalities
are also true on $\boldsymbol{T}^{n}$ if all these are tropical polynomial
functions. Thus, since

$\displaystyle\left(h\odot f_{2}^{\odot k},h\odot f_{1}^{\odot
k}\right),\left(g\odot f_{2}^{\odot k},g\odot f_{1}^{\odot
k}\right)\in\langle(f_{1},f_{2})\rangle_{\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}},$

we have

$\displaystyle\left(h\odot f_{2}^{\odot k},(g\oplus h)\odot f_{2}^{\odot
k}\right),\left(g\odot f_{2}^{\odot k},(g\oplus h)\odot f_{2}^{\odot
k}\right)\in\langle(f_{1},f_{2})\rangle_{\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}}$

by [17, Proposition 2.2(iii)]. This concludes that $\left(g\odot f_{2}^{\odot
k},h\odot f_{2}^{\odot
k}\right)\in\langle(f_{1},f_{2})\rangle_{\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}}$.
Note that if both $g$ and $h$ are tropical polynomial functions, we have also
$\left(g\odot f_{2}^{\odot k},h\odot f_{2}^{\odot
k}\right)\in\langle(f_{1},f_{2})\rangle_{\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}}$.

Let $(g,h)\in\boldsymbol{E}(V)_{1}$. By definition, we can regard that
$(g,h)\in\boldsymbol{E}(V)$, and thus there exists $k\in\boldsymbol{Z}_{>0}$
such that $\left(g\odot f_{2}^{\odot k},h\odot f_{2}^{\odot
k}\right)\in\langle(f_{1},f_{2})\rangle_{\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}}$
by the argument above if $(g,h)\not=(-\infty,-\infty)$. Clearly
$(-\infty,-\infty)=(-\infty\odot f_{2},-\infty\odot
f_{2})\in\langle(f_{1},f_{2})\rangle_{\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}}$.

For $(g,h)\in\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}^{2}$, we assume
that there exists $k\in\boldsymbol{Z}_{>0}$ such that $\left(g\odot
f_{2}^{\odot k},h\odot f_{2}^{\odot
k}\right)\in\langle(f_{1},f_{2})\rangle_{\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}}$.
By [2, Corollary 2.12], for any $x\in V$, we have $g(x)\odot f_{2}(x)^{\odot
k}=h(x)\odot f_{2}(x)^{\odot k}$. As $f_{2}(x)\in\boldsymbol{R}$, we have
$g(x)=h(x)$, and thus $(g,h)\in\boldsymbol{E}(V)_{1}$.

Finally because $f\geq 0$ on $\boldsymbol{R}^{n}$, we ahve $f_{1}\geq f_{2}$
on $\boldsymbol{R}^{n}$. By [7, Lemma 3.7.4] again, this inequality also holds
on $\boldsymbol{T}^{n}$. ∎

Now we can characterize rational function semifields of tropical curves. By
Theorem 1.1, the following lemma is clear:

######  Lemma 3.13.

Let $(x_{1},\ldots,x_{n})$ be a point of $\boldsymbol{R}^{n}$ and
$(s_{1},\ldots,s_{n})\in\boldsymbol{Z}^{n}$ a vector other than
$(0,\ldots,0)$. For $l\in\boldsymbol{R}_{\geq 0}\cup\\{\infty\\}$ and the
interval
$e_{l}:=\overline{\\{(x_{1}+ts_{1},\ldots,x_{n}+ts_{n})\,|\,t\in[0,l)\\}}$,
the congruence $\boldsymbol{E}(e_{l})$ is finitely generated as a congruence.

Let $\varGamma$ be a tropical curve. By [13, Theorem 1.1],
$\operatorname{Rat}(\varGamma)$ is finitely generated as a semifield over
$\boldsymbol{T}$. For any generators $f_{1},\ldots,f_{n}$ other than $-\infty$
of $\operatorname{Rat}(\varGamma)$ as a semifield over $\boldsymbol{T}$, by
[15, Lemma 3.11], the correspondence $X_{i}\mapsto f_{i}$ induces the
surjective $\boldsymbol{T}$-algebra homomorphism
$\psi:\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}\twoheadrightarrow\operatorname{Rat}(\varGamma)$
mapping each $X_{i}$ to $f_{i}$.

######  Corollary 3.14.

In the above setting, the kernel congruence $\operatorname{Ker}(\psi)$ is
finitely generated as a congruence.

###### Proof.

By [15, Proposition 3.12(5)], we have the isometry
$\theta:\varGamma\setminus\varGamma_{\infty}\to\boldsymbol{V}(\operatorname{Ker}(\psi));x\mapsto(f_{1}(x),\ldots,f_{n}(x))$.
Note that $\boldsymbol{V}(\operatorname{Ker}(\psi))$ has the metric defined by
the lattice length. Since $f_{1},\ldots,f_{n}$ are rational functions on
$\varGamma$, we can break $\varGamma$ into finitely many intervals
$e_{1},\ldots,e_{m}$ each which has a finite endpoint $x_{j}$ and on each
which every $f_{i}$ has a constant slope $s_{ji}$ from $x_{j}$ to the other
endpoint of $e_{j}$. Hence the image $\theta(e_{j})$ is also an interval and
is written as
$\overline{\\{(f_{1}(x_{j})+ts_{j1},\ldots,f_{n}(x_{j})+ts_{jn})\,|\,t\in[0,l(e_{j}))\\}}$,
where $l(e_{j})$ denotes the length of $e_{j}$. By Lemma 3.13 and Theorem 1.1,
we have the conclusion. ∎

Let $V$ be a closed connected subset of $\boldsymbol{R}^{n}$ of geometric
dimension one whose associated congruence $\boldsymbol{E}(V)$ is finitely
generated as a congruence. By Corollary 3.5,
$V=\boldsymbol{V}(\boldsymbol{E}(V))$ has the metric defined by the lattice
length.

######  Lemma 3.15.

If $n\geq 2$ and there exist no two distinct rays of $V$ whose primitive
direction verctors coincide, then for any ray $e$ in $V$, there exist a point
$x$ of $e$ and distinct vectors
$\boldsymbol{s}_{1},\ldots,\boldsymbol{s}_{m}\in\boldsymbol{R}^{n}\setminus\\{(0,\ldots,0)\\}$
and $f\in\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$ satisfying the
following five conditions:

$(1)$ the cone
$\\{x\\}+\operatorname{Cone}\\{\boldsymbol{s}_{1},\ldots,\boldsymbol{s}_{m}\\}$
with the vertex $x$ is of geometric dimension $n$,

$(2)$ the relative interior
$\operatorname{Int}(\\{x\\}+\operatorname{Cone}\\{\boldsymbol{s}_{1},\ldots,\boldsymbol{s}_{m}\\})$
contains $e_{x}:=\\{x+t\boldsymbol{l}\,|\,t\in\boldsymbol{R}_{\geq 0}\\}$,
where $\boldsymbol{l}$ is the primitive direction vector of $e$ such that
$e_{x}$ is contained in $e$,

$(3)$ the intersection
$(\\{x\\}+\operatorname{Cone}\\{\boldsymbol{s}_{1},\ldots,\boldsymbol{s}_{m}\\})\cap(V\setminus
e_{x})$ is empty,

$(4)$ $f$ has slope one on $e_{x}$ in the direction $\boldsymbol{l}$ with
lattice length, and

$(5)$ $f$ is constant zero on $V\setminus e_{x}$.

###### Proof.

By assumption, we can choose a point $x$ of $e$ and distinct vectors
$\boldsymbol{s}_{1},\ldots,\boldsymbol{s}_{m}\in\boldsymbol{R}^{n}\setminus\\{(0,\ldots,0)\\}$
satisfying the conditions $(1)$, $(2)$ and $(3)$.

Let $(l_{1},\ldots,l_{n}):=\boldsymbol{l}$. For any $1\leq i,j\leq n$, let
$f_{ij}$ be the element $(x)_{i}^{\odot
l_{j}}\odot(x)_{j}^{\odot(-l_{i})}\odot X_{i}^{\odot(-l_{j})}\odot
X_{j}^{\odot l_{i}}$ of $\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$. Each
$f_{\ij}$ is constant zero on $\\{x+t\boldsymbol{l}\,|\,t\in\boldsymbol{R}\\}$
and has the slope zero in the directions $\pm\boldsymbol{l}$ at any point of
$\boldsymbol{R}^{n}$. The slope of $g:=\bigoplus_{i,j=1}^{n}\left(f_{ij}\oplus
f_{ij}^{\odot(-1)}\right)$ in any direction away from the line
$\\{x+t\boldsymbol{l}\,|\,t\in\boldsymbol{R}\\}$ at any point of
$\boldsymbol{R}^{n}$ is positive. Since $f_{ij}\oplus f_{ij}^{\odot(-1)}\geq
0$, we have $g\geq 0$ and $g$ takes the value zero on and only on
$\\{x+t\boldsymbol{l}\,|\,t\in\boldsymbol{R}\\}$.

Since $\boldsymbol{l}$ is primitive, there exists
$\boldsymbol{l}^{\prime}:=(l_{1}^{\prime},\ldots,l_{n}^{\prime})\in\boldsymbol{Z}^{n}\setminus\\{(0,\ldots,0)\\}$
such that
$\boldsymbol{l}\cdot\boldsymbol{l}^{\prime}=l_{1}l_{1}^{\prime}+\cdots+l_{n}l_{n}^{\prime}=1$.
Let $h$ be the element
$(x)_{1}^{\odot(-l_{1}^{\prime})}\odot\cdots\odot(x)_{n}^{\odot(-l_{n}^{\prime})}\odot
X_{1}^{\odot l_{1}^{\prime}}\odot\cdots\odot X_{n}^{\odot
l_{n}^{\prime}}\oplus 0$ of $\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$.
Then $h$ takes the value zero on
$\\{x+y\,|\,y\in\boldsymbol{R}^{n},y\cdot\boldsymbol{l}^{\prime}\leq 0\\}$ and
$h$ has slope one in the direction $\boldsymbol{l}$ at any point of
$\\{x+y\,|\,y\in\boldsymbol{R}^{n},y\cdot\boldsymbol{l}^{\prime}\geq 0\\}$
with lattice length. Let $U_{x}(1)$ be the $1$-neighborhood of $x$, i.e.,
$U_{x}(1):=\\{y\in\boldsymbol{R}^{n}\,|\,\operatorname{dist}(x,y)=1\\}$, where
$\operatorname{dist}(x,y)=1$ means the distanse between $x$ and $y$ is one
with the usual metric. Let
$L:=\\{x+y\,|\,y\in\boldsymbol{R}^{n},y\cdot\boldsymbol{l}^{\prime}\geq
0\\}\cap U_{x}(1)$. Then, by replacing
$\boldsymbol{s}_{1},\ldots,\boldsymbol{s}_{m}$ with other vectors satisfying
the conditions $(1),(2)$ and $(3)$ if we need, we may assume that
$M:=\overline{L\setminus(\\{x\\}+\operatorname{Cone}\\{\boldsymbol{s}_{1},\ldots,\boldsymbol{s}_{m}\\})}$
is nonempty and compact. Since both $g$ and $h$ are positive on $M$, there
exists $k\in\boldsymbol{Z}_{>0}$ such that $h(y)\leq g(y)^{\odot k}$ for any
$y\in M$. For any $t\in\boldsymbol{R}_{\geq 0}$, since $h(x+t(y-x))=th(y)$ and
$f_{ij}(x+t(y-x))=tf_{ij}(y)$ for any $1\leq i,j\leq n$, we have
$h(x+t(y-x))\leq g(x+t(y-x))^{\odot k}$. Hence $f:=h\odot g^{\odot(-k)}\oplus
0\in\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$ is a desired tropical
rational function. ∎

######  Lemma 3.16.

Let $e$ be any segment of $V$ of a finite length, $x,y$ its distinct
endpoints, $z$ its midpoint and $l$ the length of $e$ with lattice length.
Then there exists $f\in\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$ such
that $f(z)=0$, the slope of $f$ is one on the segments $\overline{xz}$ and
$\overline{zy}$ with lattice length and $f$ is constant $-\frac{l}{2}$ on
$V\setminus e$.

###### Proof.

Assume that $n=1$. The tropical rational function

$\displaystyle\left\\{((-z)\odot X_{1})\oplus((-z)\odot
X_{1})^{\odot(-1)}\right\\}^{\odot(-1)}\oplus\left(-\frac{l}{2}\right)$

is desired $f$.

Assume that $n\geq 2$. Let $\boldsymbol{l}$ be the primitive direction vector
of $e$ from $x$ to $y$. For $x$ (resp. $y$), by the same argument in the proof
of Lemma 3.15, we can choose distinct vectors
$\boldsymbol{s}_{x1},\ldots,\boldsymbol{s}_{xm_{x}}\in\boldsymbol{R}^{n}\setminus\\{(0,\ldots,0)\\}$
(resp.
$\boldsymbol{s}_{y1},\ldots,\boldsymbol{s}_{ym_{y}}\in\boldsymbol{R}^{n}\setminus\\{(0,\ldots,0)\\}$)
and $f_{x}\in\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$ (resp.
$f_{y}\in\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$) satisfying the
conditions $(1)$, $(2)$ for $\boldsymbol{l}$ (resp. $-\boldsymbol{l}$) and
$(4)$ in Lemma 3.15 and $f_{x}$ (resp. $f_{y}$) is constant zero on
$V\setminus(\\{x\\}+\operatorname{Cone}\\{\boldsymbol{s}_{x1},\ldots,\boldsymbol{s}_{xm_{x}}\\})$
(resp.
$V\setminus(\\{y\\}+\operatorname{Cone}\\{\boldsymbol{s}_{y1},\ldots,\boldsymbol{s}_{ym_{y}}\\})$.
By replacing
$\boldsymbol{s}_{x1},\ldots,\boldsymbol{s}_{xm_{x}},\boldsymbol{s}_{y1},\ldots,\boldsymbol{s}_{ym_{y}}$
(and hence $f_{x}$ and $f_{y}$) if we need, we can assume that
$V\cap(\\{x\\}+\operatorname{Cone}\\{\boldsymbol{s}_{x1},\ldots,\boldsymbol{s}_{xm_{x}}\\})\cap(\\{y\\}+\operatorname{Cone}\\{\boldsymbol{s}_{y1},\ldots,\boldsymbol{s}_{ym_{y}}\\})=e$
holds. Hence the tropical rational function
$-\frac{l}{2}\odot\left(f_{x}^{\odot(-1)}\oplus
f_{y}^{\odot(-1)}\right)^{\odot(-1)}\oplus\left(-\frac{l}{2}\right)$ is a
desired one. ∎

######  Lemma 3.17.

For any $x\in V$, there exist $\varepsilon>0$, $\operatorname{val}(x)$ points
$y_{i}$ of $V$ such that the distance between $x$ and $y_{i}$ is $\varepsilon$
with lattice length and $i$ moves all integers from one to the valency
$\operatorname{val}(x)$ of $x$ in $V$, and
$f\in\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$ which has slope one in the
direction from $y_{i}$ to $x$ for any $i$ with lattice length and is constant
$-\varepsilon$ on
$V\setminus\bigcup_{i=1}^{\operatorname{val}(x)}\overline{xy_{i}}$.

###### Proof.

When $n=1$, if $x$ has valency two in $V$, then the assertion follows from
Lemma 3.16; if $x$ has valency one in $V$, then one of the tropical rational
functions $-x\odot X_{1}\oplus(-\varepsilon)$ and $x\odot
X_{1}^{\odot(-1)}\oplus(-\varepsilon)$ is desired.

Assume that $n\geq 2$. Let $\boldsymbol{l}_{i}$ be the primitive direction
vector from $x$ to $y_{i}$. For $x$ and $\boldsymbol{l}_{i}$, by the same
argument in the proof of Lemma 3.15, we can choose distinct vectors
$\boldsymbol{s}_{i1},\ldots,\boldsymbol{s}_{im_{i}}\in\boldsymbol{R}^{n}\setminus\\{(0,\ldots,0)\\}$
and $f_{i}\in\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$ satisfying the
conditions $(1)$, $(2)$ and $(4)$ in Lemma 3.15 and $f_{i}$ is constant zero
on
$V\setminus(\\{x\\}+\operatorname{Cone}\\{\boldsymbol{s}_{i1},\ldots,\boldsymbol{s}_{im_{i}}\\})$.
By replacing
$\boldsymbol{s}_{11},\ldots,\boldsymbol{s}_{1m_{1}},\boldsymbol{s}_{21},\ldots,\boldsymbol{s}_{\operatorname{val}(x)m_{\operatorname{val}(x)}}$
if we need, we can assume that if $i\not=j$, then $y_{i}$ is not in
$\\{x\\}+\operatorname{Cone}\\{\boldsymbol{s}_{j1},\ldots,\boldsymbol{s}_{jm_{j}}\\}$.
Thus for a sufficiently small positive number $\varepsilon$, the tropical
rational function
$\bigodot_{i=1}^{\operatorname{val}(x)}f_{i}^{\odot(-1)}\oplus(-\varepsilon)$
is desired. ∎

Let $V$ be a closed connected subset of $\boldsymbol{R}^{n}$ of geometric
dimension one whose associated congruence $\boldsymbol{E}(V)$ is finitely
generated as a congruence, or consist of only one point of
$\boldsymbol{R}^{n}$. Let $V^{\prime}$ be the natural compactification of $V$
as a tropical curve, i.e., let $V^{\prime}$ be obtained from $V$ by one-point
compactifications of rays of $V$.

######  Proposition 3.18.

In the above setting, $\operatorname{Rat}(V^{\prime})$ is isomorphic to
$\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}/\boldsymbol{E}(V)$ as a
$\boldsymbol{T}$-algebra if and only if there exist no two distinct rays of
$V$ whose primitive direction vectors coincide.

###### Proof.

If $V$ consists of only one point of $\boldsymbol{R}^{n}$, then
$\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}/\boldsymbol{E}(V)$ is
isomorphic to $\boldsymbol{T}=\operatorname{Rat}(V^{\prime})$ and there exist
no two distinct rays of $V$ whose primitive direction vectors coincide. If
$n=1$, then the tropical rational function on $V$ represented by $X_{1}$
generates the semifield over $\boldsymbol{T}$ isomorphic to
$\operatorname{Rat}(V^{\prime})$ as a $\boldsymbol{T}$-algebra by the proof of
[13, Proposition 3.13]. Hence we assume that $V$ does not consist of only one
point and $n\geq 2$.

We shall show the if part. Assume that there exist two distince rays
$e_{1},e_{2}$ of $V$ which have the same primitive direction vector
$(s_{1},\ldots,s_{n})\in\boldsymbol{Z}^{n}\setminus\\{(0,\ldots,0)\\}$. Let
$f$ be an element of
$\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}\setminus\\{-\infty\\}$. We can
choose
$f_{1},f_{2}\in\overline{\boldsymbol{T}[X_{1},\ldots,X_{n}]}\setminus\\{-\infty\\}$
such that $f=f_{1}\odot f_{2}^{\odot(-1)}$. Since both $f_{1}$ and $f_{2}$ are
tropical polynomial functions other than $-\infty$, we can assume that

$\displaystyle f_{1}(((x_{1})_{1}+ts_{1},\ldots,(x_{1})_{1}+ts_{n}))$
$\displaystyle=~{}$ $\displaystyle a\odot((x_{1})_{1}+ts_{1})^{\odot
i_{1}}\odot\cdots\odot((x_{1})_{n}+ts_{n})^{\odot i_{n}}$

and

$\displaystyle f_{2}(((x_{1})_{1}+ts_{1},\ldots,(x_{1})_{1}+ts_{n}))$
$\displaystyle=~{}$ $\displaystyle b\odot((x_{1})_{1}+ts_{1})^{\odot
j_{1}}\odot\cdots\odot((x_{1})_{n}+ts_{n})^{\odot j_{n}}$

with some $x_{1}\in e_{1}$, $a,b\in\boldsymbol{R}$ and
$i_{1},\ldots,i_{n},j_{1},\ldots,j_{n}\in\boldsymbol{Z}_{\geq 0}$ for any
$t\geq 0$. By the same reason, there exists $x_{2}\in e_{2}$ such that for any
$t\geq 0$,

$\displaystyle f_{1}(((x_{2})_{1}+ts_{1},\ldots,(x_{2})_{1}+ts_{n}))$
$\displaystyle=~{}$ $\displaystyle a^{\prime}\odot((x_{2})_{1}+ts_{1})^{\odot
i^{\prime}_{1}}\odot\cdots\odot((x_{2})_{n}+ts_{n})^{\odot i^{\prime}_{n}}$

and

$\displaystyle f_{2}(((x_{2})_{1}+ts_{1},\ldots,(x_{2})_{n}+ts_{n}))$
$\displaystyle=~{}$ $\displaystyle b^{\prime}\odot((x_{2})_{1}+ts_{1})^{\odot
j^{\prime}_{1}}\odot\cdots\odot((x_{2})_{n}+ts_{n})^{\odot j^{\prime}_{n}}$

with some $a^{\prime},b^{\prime}\in\boldsymbol{R}$ and
$i^{\prime}_{1},\ldots,i^{\prime}_{n},j^{\prime}_{1},\ldots,j^{\prime}_{n}\in\boldsymbol{Z}_{\geq
0}$. Then $i_{1}s_{1}+\cdots+i_{n}s_{n}$ (resp.
$j_{1}s_{1}+\cdots+j_{n}s_{n}$) must coincide with
$i^{\prime}_{1}s_{1}+\cdots+i^{\prime}_{n}s_{n}$ (resp.
$j^{\prime}_{1}s_{1}+\cdots+j^{\prime}_{n}s_{n}$). In fact, if
$i_{1}s_{1}+\cdots+i_{n}s_{n}>i^{\prime}_{1}s_{1}+\cdots+i^{\prime}_{n}s_{n}$,
then for a sufficiently large $t$, we have

$\displaystyle f_{1}(((x_{2})_{1}+ts_{1},\ldots,(x_{2})_{n}+ts_{n}))$
$\displaystyle\geq~{}$ $\displaystyle a\odot((x_{2})_{1}+ts_{1})^{\odot
i_{1}}\odot\cdots\odot((x_{2})_{1}+ts_{n})^{\odot i_{n}}$ $\displaystyle=~{}$
$\displaystyle
a+i_{1}(x_{2})_{1}+\cdots+i_{n}(x_{2})_{n}+t(i_{1}s_{1}+\cdots+i_{n}s_{n})$
$\displaystyle>~{}$ $\displaystyle
a^{\prime}+i^{\prime}_{1}(x_{2})_{1}+\cdots+i^{\prime}_{n}(x_{2})_{n}+t(i^{\prime}_{1}s_{1}+\cdots+i^{\prime}_{n}s_{n})$
$\displaystyle=~{}$ $\displaystyle
f_{1}(((x_{2})_{1}+ts_{1},\ldots,(x_{2})_{n}+ts_{n})),$

which is a contradiction. Thus $i_{1}s_{1}+\cdots+i_{n}s_{n}\leq
i^{\prime}_{1}s_{1}+\cdots+i^{\prime}_{n}s_{n}$. By the same argument, we have
$i_{1}s_{1}+\cdots+i_{n}s_{n}\geq
i^{\prime}_{1}s_{1}+\cdots+i^{\prime}_{n}s_{n}$, and hence
$i_{1}s_{1}+\cdots+i_{n}s_{n}=i^{\prime}_{1}s_{1}+\cdots+i^{\prime}_{n}s_{n}$.
Similarly, we have
$j_{1}s_{1}+\cdots+j_{n}s_{n}=j^{\prime}_{1}s_{1}+\cdots+j^{\prime}_{n}s_{n}$.
Since

$\displaystyle f(((x_{1})_{1}+ts_{1},\ldots,(x_{1})_{n}+ts_{n}))$
$\displaystyle=~{}$ $\displaystyle
f_{1}(((x_{1})_{1}+ts_{1},\ldots,(x_{1})_{n}+ts_{n}))-f_{2}(((x_{1})_{1}+ts_{1},\ldots,(x_{1})_{n}+ts_{n}))$
$\displaystyle=~{}$
$\displaystyle(a-b)+(i_{1}-j_{1})(x_{1})_{1}+\cdots+(i_{n}-j_{n})(x_{1})_{n}$
$\displaystyle+t\\{(i_{1}s_{1}+\cdots+i_{n}s_{n})-(j_{1}s_{1}+\cdots+j_{n}s_{n})\\}$
$\displaystyle=~{}$
$\displaystyle(a^{\prime}-b^{\prime})+(i^{\prime}_{1}-j^{\prime}_{1})(x_{2})_{1}+\cdots+(i^{\prime}_{n}-j^{\prime}_{n})(x_{2})_{n}$
$\displaystyle+t\\{(i_{1}s_{1}+\cdots+i_{n}s_{n})-(j_{1}s_{1}+\cdots+j_{n}s_{n})\\}$
$\displaystyle+\\{(a-b)-(a^{\prime}-b^{\prime})\\}+\\{(i_{1}-j_{1})(x_{1})_{1}-(i^{\prime}_{1}-j^{\prime}_{1})(x_{2})_{1}\\}+\cdots$
$\displaystyle+\\{(i_{n}-j_{n})(x_{1})_{n}-(i^{\prime}_{n}-j^{\prime}_{n})(x_{2})_{n}\\}$
$\displaystyle=~{}$ $\displaystyle
f_{1}(((x_{2})_{1}+ts_{1},\ldots,(x_{2})_{n}+ts_{n}))-f_{2}(((x_{2})_{1}+ts_{1},\ldots,(x_{2})_{n}+ts_{n}))$
$\displaystyle+\\{(a-b)-(a^{\prime}-b^{\prime})\\}+\\{(i_{1}-j_{1})(x_{1})_{1}-(i^{\prime}_{1}-j^{\prime}_{1})(x_{2})_{1}\\}+\cdots$
$\displaystyle+\\{(i_{n}-j_{n})(x_{1})_{n}-(i^{\prime}_{n}-j^{\prime}_{n})(x_{2})_{n}\\}$
$\displaystyle=~{}$ $\displaystyle
f(((x_{2})_{1}+ts_{1},\ldots,(x_{2})_{n}+ts_{n}))$
$\displaystyle+\\{(a-b)-(a^{\prime}-b^{\prime})\\}+\\{(i_{1}-j_{1})(x_{1})_{1}-(i^{\prime}_{1}-j^{\prime}_{1})(x_{2})_{1}\\}+\cdots$
$\displaystyle+\\{(i_{n}-j_{n})(x_{1})_{n}-(i^{\prime}_{n}-j^{\prime}_{n})(x_{2})_{n}\\},$

$f(((x_{1})_{1}+ts_{1},\ldots,(x_{1})_{n}+ts_{n}))\to\pm\infty$ as
$t\to\infty$ if and only if
$f(((x_{2})_{1}+ts_{1},\ldots,(x_{2})_{n}+ts_{n}))\to\pm\infty$ as
$t\to\infty$. If $\psi$ is a $\boldsymbol{T}$-algebra isomorphism from
$\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}/\boldsymbol{E}(V)$ to
$\operatorname{Rat}(V^{\prime})$, then
$g_{1}:=\psi(\pi_{\boldsymbol{E}(V)}(X_{1})),\ldots,g_{n}:=\psi(\pi_{\boldsymbol{E}(V)}(X_{n}))\in\operatorname{Rat}(V^{\prime})\setminus\\{-\infty\\}$
generate $\operatorname{Rat}(V^{\prime})$ as a semifield over
$\boldsymbol{T}$, where
$\pi_{\boldsymbol{E}(V)}:\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}\twoheadrightarrow\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}/\boldsymbol{E}(V)$
is the natural surjective $\boldsymbol{T}$-algebra homomorphism. By [15,
Proposition 3.12(1) and (5)], we have the isometry $\theta:V^{\prime}\setminus
V^{\prime}_{\infty}\to V;x\mapsto(g_{1}(x),\ldots,g_{n}(x))$ and
$\psi(\pi_{\boldsymbol{E}(V)}(g))(x)=g(\theta(x))$ for any
$g\in\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$ and $x\in
V^{\prime}\setminus V^{\prime}_{\infty}$. On the other hand, $V^{\prime}$ has
a rational function that has the slope one on the edge $e_{1}^{\prime}$
corresponding to $e_{1}$ and is constant on the whole of $V^{\prime}\setminus
e_{1}^{\prime}$, which is a contradiction. This means that
$\operatorname{Rat}(V^{\prime})$ is not isomorphic to
$\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}/\boldsymbol{E}(V)$ as a
$\boldsymbol{T}$-algebra.

We shall show the only if part. By Lemma 3.15, we can choose rays
$e_{1},\ldots,e_{m}$ in $V$ and
$f_{e_{1}},\ldots,f_{e_{m}}\in\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$
such that each $f_{e_{i}}$ has slope one on $e_{i}$ with lattice length and is
constant zero on $V\setminus e_{i}$ and
$\overline{V\setminus\bigcup_{i=1}^{m}e_{i}}$ is compact. Since
$\boldsymbol{E}(\overline{V\setminus\bigcup_{i=1}^{m}e_{i}})$ is finitely
generated as a congruence by Theorem 1.1 and
$\overline{V\setminus\bigcup_{i=1}^{m}e_{i}}$ is a closed connected subset of
$\boldsymbol{R}^{n}$ of geometric dimension zero or one, by Lemmas 3.16 and
3.17 and the proof of [13, Proposition 3.6], the semifield over
$\boldsymbol{T}$ generated by all
$f\in\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$ in Lemmas 3.16 and 3.17 is
isomorphic to
$\operatorname{Rat}(\overline{V\setminus\bigcup_{i=1}^{m}e_{i}})$ as a
$\boldsymbol{T}$-algebra. Note that
$\operatorname{Rat}(\overline{V\setminus\bigcup_{i=1}^{m}e_{i}})$ is the
rational function semifield of $\overline{V\setminus\bigcup_{i=1}^{m}e_{i}}$
when we regard $\overline{V\setminus\bigcup_{i=1}^{m}e_{i}}$ as a tropical
curve with lattice length. By the proof of [13, Lemma 1.4], we know that all
such $f$s and $f_{e_{1}},\ldots,f_{e_{m}}$ generate a semifield over
$\boldsymbol{T}$ isomorphic to $\operatorname{Rat}(V^{\prime})$ as a
$\boldsymbol{T}$-algebra. Since there exists a natural
$\boldsymbol{T}$-algebra homomorphism from
$\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$ to
$\operatorname{Rat}(V^{\prime})$ by regarding each element of
$\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}$ as an element of
$\operatorname{Rat}(V^{\prime})$, we have the conclusion. ∎

In the above setting, by [15, Proposition 3.12(5)], if there exists a tropical
curve $\varGamma$ such that $\operatorname{Rat}(\varGamma)$ is isomorphic to
$\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}/\boldsymbol{E}(V)$ as a
$\boldsymbol{T}$-algebra, then $\varGamma$ is isomorphic to $V^{\prime}$, and
hence $\operatorname{Rat}(V^{\prime})$ is also isomorphic to
$\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}/\boldsymbol{E}(V)$. Thus by
[13, Theorem 1.1], Corollary 3.14, [15, Proposition 3.12] and Proposition
3.18, we have the following corollary, which characterizes the rational
function semifields of tropical curves:

######  Corollary 3.19.

Let $S$ be a semifield which is finitely generated as a semifield over
$\boldsymbol{T}$. Then $S$ is isomorphic to $\operatorname{Rat}(\varGamma)$ as
a $\boldsymbol{T}$-algebra with some tropical curve $\varGamma$ if and only if
some (and hence, any) surjective $\boldsymbol{T}$-algebra homomorphism $\psi$
from a tropical rational function semifield to $S$ satisfies the following
five conditions:

$(1)$ $\operatorname{Ker}(\psi)$ is finitely generated as a congruence,

$(2)$
$\operatorname{Ker}(\psi)=\boldsymbol{E}(\boldsymbol{V}(\operatorname{Ker}(\psi)))$
holds,

$(3)$ $\boldsymbol{V}(\operatorname{Ker}(\psi))$ is connected,

$(4)$ $\boldsymbol{V}(\operatorname{Ker}(\psi))$ is of geometric dimension
zore or one, and

$(5)$ $\boldsymbol{V}(\operatorname{Ker}(\psi))$ has no two distinct rays
whose primitive direction vectors coincide.

By Corollary 3.19, [14, Corollary 1.2] and [15, Corollary 3.18], we have the
following corollary. This is a tropical analogue of the fact that the category
of nonsingular projective curves and dominant morphisms is equivalent to the
category of function fields of dimension one over $k$ and $k$-homomorphisms,
where $k$ is an algebraically closed field (see [11, 2, Corollary 6.12. in
Chapter I]).

######  Corollary 3.20.

The following categories $\mathscr{C},\mathscr{D}$ are equivalent via the
following (contravariant) functors $F,G$.

$(1)$ The class $\operatorname{Ob}(\mathscr{C})$ of objects of $\mathscr{C}$
is the tropical curves.

For $\varGamma_{1},\varGamma_{2}\in\operatorname{Ob}(\mathscr{C})$, the set
$\operatorname{Hom}_{\mathscr{C}}(\varGamma_{1},\varGamma_{2})$ of morphisms
from $\varGamma_{1}$ to $\varGamma_{2}$ consists of the surjective morphisms
$\varGamma_{1}\twoheadrightarrow\varGamma_{2}$.

$(2)$ The class $\operatorname{Ob}(\mathscr{D})$ of objects of $\mathscr{D}$
is the finitely generated semifields over $\boldsymbol{T}$ satisfying the
conditions $(1),\ldots,(5)$ in Corollary 3.19.

For $S_{1},S_{2}\in\operatorname{Ob}(\mathscr{D})$, the set
$\operatorname{Hom}_{\mathscr{D}}(S_{1},S_{2})$ of morphisms from $S_{1}$ to
$S_{2}$ consists of the injective $\boldsymbol{T}$-algebra homomorphisms
$S_{1}\hookrightarrow S_{2}$.

The functor $F:\mathscr{C}\to\mathscr{D}$ maps
$\varGamma\in\operatorname{Ob}(\mathscr{C})$ to
$\operatorname{Rat}(\varGamma)$ and for
$\varGamma_{1},\varGamma_{2}\in\operatorname{Ob}(\mathscr{C})$, maps
$\varphi\in\operatorname{Hom}_{\mathscr{C}}(\varGamma_{1},\varGamma_{2})$ to
$\varphi^{\ast}\in\operatorname{Hom}_{\mathscr{D}}(\operatorname{Rat}(\varGamma_{2}),\operatorname{Rat}(\varGamma_{1}))$.

The functor $G:\mathscr{D}\to\mathscr{C}$ maps
$S\in\operatorname{Ob}(\mathscr{D})$ to
$\boldsymbol{V}(\operatorname{Ker}(\psi_{S}))^{\prime}\in\operatorname{Ob}(\mathcal{C})$
and for $S_{1},S_{2}\in\operatorname{Ob}(\mathscr{D})$, maps
$\widetilde{\psi}\in\operatorname{Hom}_{\mathscr{D}}(S_{1},S_{2})$ to
$\varphi_{\widetilde{\psi}}\in\operatorname{Hom}_{\mathscr{C}}(\boldsymbol{V}(\operatorname{Ker}(\psi_{S_{2}}))^{\prime},\boldsymbol{V}(\operatorname{Ker}(\psi_{S_{1}}))^{\prime})$,
where $\psi_{S}$ is a fixed surjective $\boldsymbol{T}$-algebra homomorphism
from a tropical rational function semifield to $S$ and
$\boldsymbol{V}(\operatorname{Ker}(\psi_{S}))^{\prime}$ is the natural
compactification of $\boldsymbol{V}(\operatorname{Ker}(\psi_{S}))$ as a
tropical curve for any $S\in\operatorname{Ob}(\mathscr{D})$ and
$\varphi_{\widetilde{\psi}}$ is the unique surjective morphism
$\boldsymbol{V}(\operatorname{Ker}(\psi_{S_{2}}))^{\prime}\twoheadrightarrow\boldsymbol{V}(\operatorname{Ker}(\psi_{S_{1}}))^{\prime}$
such that $\widetilde{\psi}=\left(\varphi_{\widetilde{\psi}}\right)^{\ast}$
given in [14, Theorem 1.1] and [15, Corollary 3.18].

Let $\varGamma$ be a tropical curve and $D$ an effective divisor on
$\varGamma$. Let $F:=\\{f_{1},\ldots,f_{n}\\}$ be a finite generating set of
$R(D)$ as a $\boldsymbol{T}$-module such that $f_{i}\not=-\infty$ for all $i$
and
$\phi_{F}:\varGamma\setminus\varGamma_{\infty}\to\boldsymbol{R}^{n};x\mapsto(f_{1}(x),\cdots,f_{n}(x))$
the rational map induced by $F$. Note that $F$ contains all extremals of
$R(D)$ up to tropical scalar multiplication (other than $-\infty$).

######  Corollary 3.21.

In the above setting, if $\phi_{F}$ is injective, then
$\operatorname{Rat}(\varGamma)$ is isomorphic to
$\overline{\boldsymbol{T}(X_{1},\ldots,X_{n})}/\boldsymbol{E}(\operatorname{Im}(\phi_{F}))$
as a $\boldsymbol{T}$-algebra. In particular, if $\operatorname{deg}(D)\geq
2g(\varGamma)+1$, then $\phi_{F}$ is injective.

###### Proof.

If $\varGamma$ is a singleton, then the assertion is clear. Assume that
$\varGamma$ is not a singleton. By the definition of $\phi_{F}$ and Theorem
1.1, $\operatorname{Im}(\phi_{F})$ is a closed subset of $\boldsymbol{R}^{n}$
of geometric dimension one whose associated congruence
$\boldsymbol{E}(\operatorname{Im}(\phi_{F}))$ is finitely generated as a
congruence. By [12, Lemma 3.3.10], if $\phi_{F}$ is injective, then $\phi_{F}$
is a local isometry on the image. If $F$ consists of only extremals, i.e., $F$
is minimal, and the statement holds, then it is also true when $F$ is an
arbitrary finite generating set of $R(D)$ since $\operatorname{Im}(\phi_{F})$
cannot have two distinct rays whose primitive direction vectors coincide.
Hence, without loss of generality, we may assume that $F$ has only extremals.
Assume that there exist two distinct rays $e_{1}$ and $e_{2}$ of
$\operatorname{Im}(\phi_{F})$ which have the same primitive direction vector
$(s_{1},\ldots,s_{n})\in\boldsymbol{Z}^{n}\setminus\\{(0,\ldots,0)\\}$. Let
$e_{i}^{\prime}$ be the (closed) edge of $\varGamma$ corresponding to $e_{i}$
and $x_{i}^{\prime}$ the endpoint of $e_{i}^{\prime}$ at infinity. By
assumption, for any $j$, the slopes of $f_{j}$ on $e_{1}^{\prime}$ and
$e_{2}^{\prime}$ in the direction to $x_{1}^{\prime}$ and $x_{2}^{\prime}$ are
$s_{j}$. By [12, Lemma 3.3.10] again, there exists at least one $j$ such that
$s_{j}$ is one or minus one. In particular, if $\varGamma$ is a tree, there
exists $j$ such that $s_{j}=-1$. Then we have
$(D+\operatorname{div}(f_{j}))(x_{1}^{\prime})=D(x_{1}^{\prime})+\operatorname{div}(f_{j})(x_{1}^{\prime})>0$
and for a rational function $f$ on $\varGamma$ which has slope one from
$x_{1}^{\prime}$ to $x_{2}^{\prime}$ and is constant elsewhere, the rational
function $f\odot f_{j}$ is an extremal of $R(D)$ by [10, Lemma 3.11] and this
has different slopes on $e_{1}^{\prime}$ and $e_{2}^{\prime}$ in the
directions to $x_{1}^{\prime}$ and $x_{2}^{\prime}$, respectively. This is a
contradiction.

Assume that $\varGamma$ is not a tree. Let $x_{0}^{\prime}$ be the nearest
point to $x_{1}^{\prime}$ whose valency is at least three. We first consider
the case $s_{j}=-1$. Then we have
$D(x_{1}^{\prime})+\operatorname{div}(f_{j})(x_{1}^{\prime})>0$ again and a
rational function $g$ on $\varGamma$ which has slope one from $x_{1}^{\prime}$
to $x_{0}^{\prime}$, the rational function $g\odot f_{j}$ is an extremal of
$R(D)$ by [10, Lemma 3.11] and this has different slopes on $e_{1}^{\prime}$
and $e_{2}^{\prime}$ in the directions to $x_{1}^{\prime}$ and
$x_{2}^{\prime}$, respectively, which is a contradiction. We next consider the
case $s_{j}=1$. Let $T$ be the maximum subgraph of $\varGamma$ which is a tree
and contains $x_{1}^{\prime}$. Then there exists a rational function $h\in
R(D+\operatorname{div}(f_{j}))$ such that
$D+\operatorname{div}(f_{j})+\operatorname{div}(h)$ is zero on
$T\setminus\\{x_{1}^{\prime}\\}$ and $h$ is constant on $\varGamma\setminus
T$. Since $D(x_{1}^{\prime})+\operatorname{div}(f_{j})(x_{1}^{\prime})>0$, the
rational function $g^{\odot(-1)}\odot h\odot f_{j}$ is an extremal of $R(D)$
and has different slopes on $e_{1}^{\prime}$ and $e_{2}^{\prime}$ in the
dirextions to $x_{1}^{\prime}$ and $x_{2}^{\prime}$, respectively. It is a
contradiction.

Therefore, if $\phi_{F}$ is injective, then there exist no distinct rays on
$\operatorname{Im}(\phi_{F})$ which have the same primitive direction vector,
and hence we have the conclusion by Proposition 3.18. The last assertion
follows from [10, Theorem 50 and Lemma 41]. ∎

## References

  * [1] George Bergman, The logarithmic limit-set of an algebraic variety, Transactions of the American Mathematical Society 157:459–469, 1971.
  * [2] Aaron Bertram and Robert Easton, The tropical Nullstellensatz for congruences, Advances in Mathematics 308:36–82, 2017.
  * [3] Melody Chan, Tropical hyperelliptic curves, Journal of Algebraic Combinatorics 37:331-359, 2013.
  * [4] Filip Cools, Jan Draisma, Sam Payne and Elina Robeva, A tropical proof of the Brill–Noether theorem, Advances in Mathematics 230(2):759–776, 2012.
  * [5] Israel M. Geldand, Mikhail M. Kapranov and Andrei V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser Boston, 1994.
  * [6] Jeffrey Giansiracusa and Noah Giansiracusa, Equations of tropical varieties, Duke Mathematical Journal 165(18):3379–3433, 2016.
  * [7] Jeffrey Giansiracusa and Noah Giansiracusa, The universal tropicalization and the Berkovich analytification, Kybemetika 58(5):790–815, 2022.
  * [8] Jonathan S. Golan, Semirings and their applications, Updated and expanded version of The theory of semirings, with applications to mathematics and theoretical computer science. Kluwer Academic Publishers, Dordrecht, 1999.
  * [9] Dima Grigoriev, A tropical version of Hilbert function, arXiv:2404.06440.
  * [10] Christian Haase, Gregg Musiker and Josephine Yu, Linear Systems on Tropical Curves, Mathematische Zeitschrift 270, no. 3-4, 1111–1140, 2012.
  * [11] Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.
  * [12] Song JuAe, Galois quotients of tropical curves and invariant linear systems, Hokkaido Mathematical Journal 51(3):445-486, 2022.
  * [13] Song JuAe, Rational function semifields of tropical curves are finitely generated over the tropical semifield, International Journal of Algebra and Computation 32(08):1575–1594, 2022.
  * [14] Song JuAe, $\boldsymbol{T}$-algebra homomorphisms between rational function semifields of tropical curves, arXiv:2304.03508.
  * [15] Song JuAe, Congruences on tropical rational function semifields of tropical curves, arXiv:2305.04204v2.
  * [16] Song JuAe, Semiring isomorphisms between rational function semifields of tropical curves, Journal of Pure and Applied Algebra, 228(9), 107683, 2024.
  * [17] Dániel Joó and Kalina Mincheva, Prime congruences of additively idempotent semirings and a Nullstellensatz for tropical polynomials, Selecta Mathematica 24(3):2207–2233, 2018.
  * [18] Dániel Joó and Kalina Mincheva, On the dimension of polynomial semirings Journal of Algebra 507:103-119, 2018.
  * [19] Masanori Kobayashi and Shinsuke Odagiri, Tropical geometry of PERT, Journal of Math-for-Industry 5(B):145–149, 2013.
  * [20] Diane Maclagan and Bernd Sturmfels, Introduction to tropical geometry, Graduate Studies in Mathematics, Vol. 161. American Mathematical Soc., Providence, RI, 2015.
  * [21] Grigory Mikhalkin, Enumerative Tropical Algebraic Geometry in $\boldsymbol{R}^{2}$, Journal of the American Mathematical Society 18(2):313–377, 2005.
  * [22] Imre Simon, Recobnizable Sets with Multiplicities in the Tropical Semiring, International Symposium on Mathematical Foundations of Computer Science. Berlin, Heidelberg: Springer Berlin Heidelberg, 1988.
  * [23] Bernd Sturmfels, Solving Systems of polynomial equations, Vol. 97. American Mathematical Society, 2002.
  * [24] Oleg Viro, Patchworking real algebraic varieties, arXiv:0611382.