Given a tropical divisor $D$ in the intersection of two tropical plane curves, we study when it can be realized as the tropicalization of the intersection of two algebraic curves, and give a sufficient condition.

We show that under a certain condition involving a graph determined by these tropical curves, we can algorithmically find algebraic curves such that the tropicalization of their intersection is $D$.

§ INTRODUCTION

In this paper, let $k$ be a fixed algebraically closed field with a nontrivial valuation $\val: k\to \mathbb{R}\cup \{+\infty\}$.

A tropical plane curve is obtained by the tropicalization of an algebraic curve in $(k^*)^2$.
Here, the tropicalization is defined using the following map:

\begin{eqnarray*}
\trop: (k^*)^2&\to&{\mathbb{R}}^2\\
(x, y)&\mapsto&(-\val(x), -\val(y)).
\end{eqnarray*}

Let $f=\sum_{ij}c_{ij}x^iy^j, g=\sum_{ij}d_{ij}x^iy^j \in k[x^{\pm1}, y^{\pm1}]$ be given.
For a given tropical rational function $\psi$ defined on the tropical plane curve $\Trop(V(f))$, it has been considered whether the zero points of $\psi$ can be obtained by the tropicalization of the intersection of two algebraic curves ([1], [5], [8], [9] and [10]).
This kind of problem is called a tropical lifting problem or a tropical realization problem.
In this paper, we give a sufficient condition involving a graph determined by given tropical curves for the lifting problem for the intersection of curves.

§.§ Tropical lifting problem

First, we explain what is known about tropical lifting problems for the intersection of two tropical plane curves.
Let $\cF$ and $\cG$ be bivariate tropical polynomials.
They define the tropical curves $V(\cF)$ and $V(\cG)$ (see Section 2).

We say that two tropical plane curves $\Gamma_1$ and $\Gamma_2$ meet properly at a point $p$ if $p$ is an isolated point in $\Gamma_1\cap \Gamma_2$.
We define $\cPI(\cF, \cG)$ as the multiset of the points $p$ at which $V(\cF)$ and $V(\cG)$ meet properly, with the local intersection numbers as multiplicities.
We also write $\cPI(\trop(f), \trop(g))$ as $\cPI(f, g)$.

Proper intersections are the simplest intersections of tropical plane curves.
Tropical lifting problems of proper intersections are studied in [10] (see Theorem <ref>).
For algebraic curves $C_1, C_2\subset (k^*)^2$, if the tropical curves $\Trop(C_1)$ and $\Trop(C_2)$ meet properly, then $\trop(C_1\cap C_2)$ is equal to the intersection $\Trop(C_1)\cap \Trop(C_2)$, considered with multiplicities.
Thus, we have to consider the case where $\Trop(C_1)\cap \Trop(C_2)$ does not consist of isolated points, i.e., contains $1$-dimensional components.

A divisor on a tropical curve $\Gamma$ is a finite sum $D=\sum n_iP_i$, where $P_i\in \Gamma$ and $n_i\in \mathbb{Z}$.

A tropical rational function on a tropical curve $\Gamma$ is a continuous function $\psi: \Gamma\to \mathbb{R}$ such that its restriction to any edge of $\Gamma$ is a piecewise linear function with integer slopes, i.e., piecewise $\mathbb{Z}$-affine, and with only finitely many pieces.
The divisor of $\psi$ is $\sum_{P\in \Gamma}\ord_P(\psi)P$, where $\ord_P(\psi)$ is $(-1)$ times the sum of the outgoing slopes of $\psi$ at $P$.
We write $(\psi)$ for the divisor of $\psi$.
If $D$ and $E$ are divisors such that $D-E=(\psi)$ for some tropical rational function $\psi$, we say that $D$ and $E$ are linearly equivalent.
We define the support of $\psi$ as $\Supp(\psi)=\overline{\{P\in \Gamma\ |\ \psi(P)\neq 0\}}$.

Morrison showed the following necessary condition for the realizability of zero points of a tropical rational function.

Let $\Gamma_1$ and $\Gamma_2$ be tropical plane curves.
Let $E$ be the stable intersection divisor (Definition <ref>) of $\Gamma_1$ and $\Gamma_2$, and let $D=\sum n_iP_i\ (n_i\in \mathbb{Z}_{\geq 0})$ be a divisor on $\Gamma_1\cap \Gamma_2$ such that each coordinate of $P_i$ is in the value group.
Assume that there exist algebraic curves $C_1, C_2\subset (k^*)^2$ without common irreducible components such that $\Trop(C_1)=\Gamma_1$, $\Trop(C_2)=\Gamma_2$, and  $\trop(C_1\cap C_2)=D$ as multisets.
Then, there exists a tropical rational function $\psi$ on $\Gamma_1$ such that $(\psi)=D-E$ and $\Supp(\psi)\subset \Gamma_1 \cap \Gamma_2$.

In [8], a conjecture on the converse is also presented.

Let $\psi$ be a tropical rational function on a tropical curve $\Trop(V(f))$ such that $\Supp(\psi)\subset \Trop(V(f)) \cap \Trop(V(g))$ and $(\psi)=D-E$, where $E$ is the stable intersection divisor and $D=\sum n_iP_i\ (n_i\in \mathbb{Z}_{\geq 0})$ is a divisor on $\Gamma_1\cap \Gamma_2$ such that each coordinate of $P_i$ is in the value group.
Then is it possible to find $f', g'\in k[x^{\pm1}, y^{\pm1}]$ such that $\Trop(V(f'))=\Trop(V(f))$, $\Trop(V(g'))=\Trop(V(g))$ and $\trop(V(f', g'))=D$?

This was answered in the negative.
For example, see <cit.> for a self-intersection case, and <cit.> for a non self-intersection case.
On the other hand, it would be useful to find sufficient conditions for the realizability.
The purpose of this paper is to give a sufficient condition involving a certain graph.
We introduce several notations before explaining the setting of the main problem.

Let $\Gamma_1$ and $\Gamma_2$ be tropical plane curves.
Let $C$ be a connected component of $\Gamma_1 \cap \Gamma_2$.
The intersection multiplicity of $\Gamma_1 \cap \Gamma_2$ on $C$ is defined as the sum of the multiplicities of the stable intersection points on $C$ (Definitions <ref> and <ref>).

Let us introduce notations on the second simplest components of the intersection of tropical curves.

We define $\cRayFG$ as the set of rays $L$ satisfying the following:

* $L$ is a connected component of the intersection $V(\cF) \cap V(\cG)$.

* The intersection multiplicity of $V(\cF)$ and $V(\cG)$ on $L$ is  $1$.

Also, we define $\cLSFG$ as the set of (bounded) line segments $L$ satisfying the following:

* $L$ is a connected component of the intersection $V(\cF) \cap V(\cG)$.

* The intersection multiplicity of $V(\cF)$ and $V(\cG)$ on $L$ is  $2$.

We write $\cSIFG:=\cRayFG \cup \cLSFG$.
We also write $\cRay(\trop(f), \trop(g))$, $\cLS(\trop(f), \trop(g))$ and $\cSI(\trop(f), \trop(g))$ as $\cRayfg$, $\cLSfg$ and $\cSIfg$, respectively.
For a description of $V(\cF)$ and $V(\cG)$ near a ray or a line segment $L\in \cSIFG$, see Lemma <ref>.

Thus, the connected components of $V(\cF)\cap V(\cG)$ are points in $\cPI(\cF, \cG)$, elements of $\cSIFG$, and possibly a number of other $1$-dimensional sets.

We will see that, if $L\in \cRayfg$, then there are at most one point in the intersection $\trop(V(f, g))\cap L$ (Corollary <ref>).
Thus, in this paper, we will consider a divisor $D$ on $\Trop(V(f)) \cap \Trop(V(g))$ satisfying the following condition:

$(*)$ For $D=\sum n_iP_i$, we have $n_i\geq 0$ and each coordinate of $P_i$ is in the value group.
There exists a tropical rational function $\psi$ on the tropical curve $\Trop(V(f))$ such that $\Supp(\psi)\subset \Trop(V(f)) \cap \Trop(V(g))$ and $(\psi)=D-E$, where $E$ is the stable intersection divisor of $\Trop(V(f))$ and $\Trop(V(g))$.
If $L\in \cRayfg$, then we require $|D|_L|=1$.

Note that this condition is natural in view of Theorem <ref>.

For a tropical plane curve $\Gamma$, we write $\Sn(\Gamma)$ for the set of the $n$-dimensional cells of $\Gamma$ (see Theorem <ref>).
For a tropical polynomial $\cF \in \mathbb{T}[x^{\pm1}, y^{\pm1}]$, we write $\Delta^{(n)}_{\cF}$ for the set of the $n$-dimensional cells of $\Delta_{\cF}$, where $\Delta_{\cF}$ is the dual subdivision of the support of $\cF$ (see Definition <ref>).
For a Laurent polynomial $h\in k[x^{\pm1}, y^{\pm1}]$, we write $\Delta_h$ and $\Delta^{(n)}_h$ for $\Delta_{\trop(h)}$ and $\Delta_{\trop(h)}^{(n)}$, respectively.

Note that the intersection multiplicity at an endpoint of a ray or a line segment $L\in \cSIfg$ must be at least $1$, and hence the tropical curves  $\Trop(V(f))$ and $\Trop(V(g))$ have no vertices in the interior of $L$ (see Lemma <ref> for details).
Thus, we can define the following maps.

We define maps $\phi_i$ ($i=1, 2$) as follows:

\begin{eqnarray*}
\phi_1: \cSIFG&\to&\Sone(V(\cF))\\
L&\mapsto&\text{the $1$-dimensional cell of $V(\cF)$ containing $L$},\\
\phi_2: \cSIFG&\to&\Sone(V(\cG))\\
L&\mapsto&\text{the $1$-dimensional cell of $V(\cG)$ containing $L$},
\end{eqnarray*}

and we define maps $\Phi_i$ ($i=1, 2$) as follows:

\begin{eqnarray*}
\Phi_1: \cSIFG&\to&\Delta_{\cF}^{(1)}\\
L&\mapsto&\text{the $1$-simplex of $\Delta_{\cF}$ corresponding to $\phi_1(L)$},\\
\Phi_2: \cSIFG&\to&\Delta_{\cG}^{(1)}\\
L&\mapsto&\text{the $1$-simplex of $\Delta_{\cG}$ corresponding to $\phi_2(L)$}.
\end{eqnarray*}

Let $a, b\in \mathbb{R}^2$ ($a\neq b$) be points such that the line segment $\sembunab$ has a rational slope.
Then, there is a primitive integer vector $v\in \mathbb{Z}^2$ which has the same slope as $\sembunab$.
The lattice length of $\sembunab$ is the ordinary length of $\sembunab$ divided by the ordinary length of $v$.
When $a=b$, we define the lattice length of $\sembunab$ as $0$.
We write $\dist(a, b)$ for the lattice length of $\sembunab$.
We note that $\dist$ does not satisfy the metric inequality.

On a line segment $L\in \cLS(\cF, \cG)$, a divisor $D$ satisfying $(*)$ can be described as follows.

Let $L\in \cLSFG$ be a line segment.
Let $\psi$ be a tropical rational function on $V(\cF)$ such that there is a neighborhood $U$ of $L$ satisfying $U\cap \Supp(\psi)\subset L$, that it takes zero at the endpoints $P_+, P_-$ of $L$ and that $(\psi)|_L=D|_L-(P_++P_-)$, where $D|_L$ is the restriction of $D$ to $L$.
Then, $D|_L$ consists of two points $P_1$ and $P_2$, and we have $\dist(P_+, P_1)= \dist(P_-, P_2)$.

Let $D$ be a divisor satisfying $(*)$ and $(\psi)=D-E$.

For $L\in \cLSfg$, we define $\dist(D|_L, E|_L)=\min\{\dist(P_+, P_1), \dist(P_+, P_2)\}$ using the notation of Lemma <ref>.
Also, when $L\in \cRay(f, g)$, we write $\dist(D|_L, E|_L)$ for the lattice length of the distance of the point in $D|_L$ and the endpoint of $L$.
In this case, $\dist(D|_L, E|_L)$ may be $\infty$.

By analogy with plane algebraic curves, it is natural to fix $f$ and change $g$ in realizing $D$, i.e., the zeros of $\psi$.
For example, in [5], a tropical curve $\Gamma$ and an algebraic curve $C$ satisfying $\Trop(C)=\Gamma$ are fixed and $\trop(C\cap C')$ are studied for curves $C'$ with $\Trop(C')=\Gamma$.
Also, it seems to be useful to study whether it is possible to realize a certain part of $D$.
Let $\cSI'$ be a subset of $\cSI(f, g)$ and $\cPI:=\cPI(f, g)$.
Let $D|_{\cSI'\cup \cPI}$ denote the restriction of $D$ to the union of the elements of $\cSI'\cup \cPI$.
Then, is it possible to realize $D|_{\cSI'\cup \cPI}$, i.e., does there exist a Laurent polynomial $g'\in k[x^{\pm1}, y^{\pm1}]$ such that $\Trop(V(g'))=\Trop(V(g))$ and $\trop(V(f, g'))|_{\cSI' \cup \cPI}=D|_{\cSI'\cup \cPI}$?

§.§ Main result

Our main theorems give sufficient conditions for the realizability, and a partial answer to the above question.
To state the main theorems, we introduce terminologies on trees.

It is well known that any two vertices of a tree $T$ are connected by a unique simple path in $T$ (see <cit.>).
We write $pTq$ for the simple path between two vertices $p$ and $q$ in $T$.

Let $T$ be a tree and $\leq$ a total ordering on the set of its vertices.
Let $p_0$ denote the smallest vertex for $\leq$.
The order $\leq$ is called normal if $p\in p_0Tq$ implies $p\leq q$.

For lattice points $\bi, \bj \in \mathbb{Z}^2$ such that $\bj-\bi$ is primitive and a tropical polynomial $\mathcal{F}=\bigoplus_{\mathbf{i}\in \mathbb{Z}^2} \alpha_{\mathbf{i}} \mathbf{x}^{\bi}$ with $\alpha_{\bi}, \alpha_{\bj}\neq -\infty$, where $\mathbf{x}^{(i_1, i_2)}$ denotes $x^{i_1}y^{i_2}$, we define $\mu_n(\cF; \sembunij)$ ($n\in \mathbb{Z}$) and $\mu(\cF; \sembunij)$ by

\begin{eqnarray*}
\mu_n(\cF; \sembunij)&:=&\alpha_{\bi+n(\bj-\bi)}-\alpha_{\bi}-n(\alpha_{\bj}-\alpha_{\bi}),\\
\mu(\cF; \sembunij)&:=&\min\{\mu_n(\cF; \sembunij)\ |\ n\in \mathbb{Z}\setminus \{0, 1\}\}.
\end{eqnarray*}

For $f=\sum_{\bi}c_{\bi}\mathbf{x}^{\bi}\in k[x^{\pm1}, y^{\pm1}]$ with $c_{\bi}, c_{\bj} \neq 0$, we write $\mu_n(f; \sembunij)$ and $\mu(f; \sembunij)$ for $\mu_n(\trop(f); \sembunij)$ and $\mu(\trop(f); \sembunij)$, respectively.

Note that $\mu_0(\cF; \sembunij)=\mu_1(\cF; \sembunij)=0$.

Let $\mathcal{F}=\bigoplus_{\mathbf{i}\in \mathbb{Z}^2} \alpha_{\mathbf{i}} \mathbf{x}^{\bi}$ be a tropical polynomial, $\sembunij$ a $1$-simplex of $\Delta_{\cF}$ and $L$ the corresponding edge of $V(\cF)$ (see Theorem <ref>).
Then, for any $P\in L$ and $n\in \mathbb{Z}\setminus \{0, 1\}$, we have the following (see Remark <ref>):

\[
\alpha_{\mathbf{i}}+\mathbf{i}\cdot P=\alpha_{\mathbf{j}}+\mathbf{j}\cdot P>\alpha_{\mathbf{i}+n(\mathbf{j}-\mathbf{i})}+(\mathbf{i}+n(\mathbf{j}-\mathbf{i}))\cdot P,
\]

and hence,

\[
\alpha_{\mathbf{i}+n(\mathbf{j}-\mathbf{i})}-\alpha_{\mathbf{i}}-n(\alpha_{\mathbf{j}}-\alpha_{\mathbf{i}})>0,
\]

\[
\mu_n(\cF; \sembunij)>0.
\]

Thus, in this case, we have $\mu(g; \sembunij)>0$.
In particular, if $L\in \cSIfg$, $P\in L$ and $\sembunij=\Phi_2(L)$, then $\sembunij$ a $1$-simplex of $\Delta_{g}$, and hence, $\mu_n(g; \sembunij)>0$.

The value $\mu(\cF; \sembunij)$ measures the margin for $\sembunij$ to be a $1$-simplex of $\Delta_{\cF}$, in a sense.

Now, let us state the main theorems.
We consider the following graph theoretic condition which will be crucial in our sufficient conditions.

We say that $\cSI'$ is acyclic with respect to $\Phi_2$, if the map $\Phi_2|_{\cSI'}$ is injective, i.e., there is no duplication in $\Delta':=\Phi_2(\cSI')$, and the union of the elements of $\Delta'$ is a forest.

The following theorem implies that $D$ can be realized on $\cSI' \cup \cPI$ if $\cSI'$ is acyclic with respect to $\Phi_2$ and $D$ is sufficiently close to $E$.

(=Theorem <ref>)
Let $D$ be a divisor on $\Trop(V(f))$ satisfying $(*)$ and $(\psi)=D-E$.
Assume that $\cSI'$ is acyclic with respect to $\Phi_2$ and that for each $L\in \cSI'$, we have $\dist(D|_L, E|_L)<\mu(g; \sembunij)$ $(\sembunij:=\Phi_2(L))$.
Then, there exists $g'\in k[x^{\pm1}, y^{\pm1}]$ such that $\trop(g')=\trop(g)$ and

\[
\trop(V(f, g'))|_{\cSI' \cup \cPI}=D|_{\cSI' \cup \cPI}.
\]

For a given set $S\subset \mathbb{R}^n$, we write $\Aff(S)$ for the affine span of $S$.

(=Theorem <ref>)
Let $D$ be a divisor on $\Trop(V(f))$ satisfying $(*)$ and $(\psi)=D-E$.
Assume that $\cSI'$ is acyclic with respect to $\Phi_2$ and that we can number and order the endpoints of the elements of $\Delta'$ as $p_1< \dots < p_n$ so that this order is normal on each tree of the forest.
Assume that for each element $\sembunpipj$ of $\Delta'$, its affine span $\Aff(\sembunpipj)$ satisfies the following conditions:

* It does not contain other elements of $\Delta'$.

* It does not contain a point $p_l$ with $l>i, j$.

Then, there exists $g'\in k[x^{\pm1}, y^{\pm1}]$ such that $\trop(g')=\trop(g)$ and

\[
\trop(V(f, g'))|_{\cSI' \cup \cPI}=D|_{\cSI' \cup \cPI}.
\]

The proofs of the theorems proceed as follows.
For an element $L\in \cSIfg$, we will give an algorithm to determine $\trop(V(f)\cap V(g))\cap L$.
This algorithm proceeds by making a suitable Laurent polynomial from the ideal $(f, g)$ and allows us to find a desired Laurent polynomial $g'$.

We will determine the coefficients of this Laurent polynomial one by one.
In the setting of Theorem <ref>, we use a given ordering.
We need the acyclicity condition to maintain the consistency.

Let $L\in \cSI'$ and $\Phi_2(L)=\sembunpipj$.
When we determine the coefficient $d'_{p_{i}}$ of $g'$, we consider the coefficients $d_{\mathbf{i}}$ ($\mathbf{i}\in \Aff(\sembunpipj)$).
This is why Theorems <ref> and <ref> require the condition about the coefficients $d_{\mathbf{i}}$ for $\mathbf{i}\in \Aff(\sembunpipj)$.

The condition $p_l\notin \Aff(\sembunpipj)$ $(l>i, j)$ depends on the ordering, not just on $\cSI'$.
For example, the order on the left of Figure <ref> satisfies the condition, but the one on the right does not.

[label=below:$p_3$] (L1) at (0,0);
[label=below:$p_4$] (L2) at (0.5,0);
[label=below:$p_1$] (L3) at (1,0);
[label=above:$p_2$] (L4) at (0.5,0.5);
[label=below:$p_2$] (L5) at (2,0);
[label=below:$p_1$] (L6) at (2.5,0);
[label=below:$p_4$] (L7) at (3,0);
[label=above:$p_3$] (L8) at (2.5,0.5);

[very thick] (L2)–(L1)–(L4)–(L3);
[very thick] (L6)–(L5)–(L8)–(L7);

in 1,2,...,8 [black] (L) circle (0.05);

Two orderings of the endpoints of the elements of $\Delta'=\Phi_2(\cSI')$.

The rest of this paper is organized as follows.
Section 2 gives fundamental definitions and properties about tropical curves.
In Section 3, we show several lemmas concerning properties of $\cF$ and $\cG$ in a neighborhood of $L\in \cSIfg$ and a kind of division for Laurent polynomials over a valuation field.
In Section 4, we give an algorithm to find the intersection of $\trop(V(f, g))$ and $L\in \cSIfg$, and prove the main theorems.
In the last Section 5, we give several examples concerning the main theorems to illustrate the necessity of the acyclicity condition.

§ ACKNOWLEDGEMENTS

I am grateful to Nobuyoshi Takahashi for helpful comments.

§ TROPICAL CURVES

In this section, we recall the basics about tropical plane curves.

First we give fundamental definitions and facts oin tropical geometry.
For details, see [6].

We give the definition of the tropical algebra which is essential for studying tropical geometry.

We define $\mathbb{T}=\mathbb{R}\cup \{-\infty\}$.
The tropical algebra is the triple $(\mathbb{T}, \oplus, \odot)$, where the addition $\oplus$ is defined as the operation that takes the maximum of two numbers and the multiplication $\odot$ is defined as the ordinary addition.
We can easily check that $(\mathbb{T}, \oplus, \odot)$ is a semifield.

To define tropical plane curves in terms of tropical algebra, we define tropical polynomials.

A tropical polynomial $\mathcal{F}$ is an expression of the form

\[
\mathcal{F}=\bigoplus_{\mathbf{i}} \alpha_{\mathbf{i}} x_1^{i_1}\dots x_n^{i_n},
\]

where $\mathbf{i}=(i_1, \dots, i_n)\in \mathbb{Z}^n$ and $\alpha_{\mathbf{i}}\in \mathbb{T}$, and only finitely many of the coefficients $\alpha_{\mathbf{i}}$ are $-\infty$.
We may drop terms with coefficients $-\infty$.
A tropical polynomial defines a map from $\mathbb{R}^n$ to $\mathbb{R}\cup \{-\infty\}$ in a natural way:

\[
\mathcal{F}(t_1, \dots, t_n)=\max_{\mathbf{i}}(\alpha_{\mathbf{i}}+i_1t_1+\dots+i_nt_n).
\]

We write $\mathbb{T}[x^{\pm1}, y^{\pm1}]$ for the set of all bivariate tropical polynomials.

Let $\mathcal{F}=\bigoplus_{\mathbf{i}} \alpha_{\mathbf{i}} x_1^{i_1}\dots x_n^{i_n}\neq -\infty$ be a tropical polynomial.
The tropical hypersurface $V(\mathcal{F})$ defined by $\mathcal{F}$ is the set

\begin{eqnarray*}
V(\mathcal{F})=\left\{ (t_1,\dots, t_n)\in \mathbb{R}^n \middle|
\begin{array}{l}
\text{$\exists \mathbf{i}=(i_1, \dots, i_n), \mathbf{j}=(j_1, \dots, j_n) \in \mathbb{Z}^n$ ($\mathbf{i} \neq \mathbf{j}$) s.t.}\\
\text{$\alpha_{\mathbf{i}}+i_1t_1+\dots+i_nt_n=\alpha_{\mathbf{j}}+j_1t_1+\dots+j_nt_n$}\\
\hspace{33.5mm} =\mathcal{F}(t_1,\dots, t_n)
\end{array}
\right\}.
\end{eqnarray*}

If $\mathcal{F}=-\infty$, i.e. all the coefficients of $\mathcal{F}$ are $-\infty$, we define $V(-\infty)=\mathbb{R}^n$.
When $n=2$ and $\mathcal{F}\neq -\infty$, we call $V(\mathcal{F})$ a tropical plane curve.
Note that we will consider a tropical plane curve as a polyhedral complex endowed with multiplicities on its $1$-dimensional cells (see Definition <ref>).

The following map is a bridge between algebraic geometry and tropical geometry.

We define the tropicalization map as follows:

\begin{eqnarray*}
\trop: &(k^*)^n&\to{\mathbb{R}}^n\\
&(x_1, \dots, x_n)&\mapsto (-\val(x_1), \dots, -\val(x_n)).
\end{eqnarray*}

For $A\subset (k^*)^n$, we write $\Trop(A)$ for the closure of $\trop(A)$ in $\mathbb{R}^n$.

Recall that $k$ is an algebraically closed field with a nontrivial valuation.

Let $f=\sum_{\mathbf{i}} c_{\mathbf{i}} \mathbf{x}^{\bi}\in k[x_1^{\pm1},\dots, x_n^{\pm1}]$ be a Laurent polynomial.
We define the tropicalization of $f$ as

\[
\trop(f)=\bigoplus_{\mathbf{i}} \trop(c_{\mathbf{i}}) \mathbf{x}^{\bi}\ (=\bigoplus_{\mathbf{i}} (-\val(c_{\mathbf{i}}) \mathbf{x}^{\bi})).
\]

Let $f\in k[x_1^{\pm1},\dots, x_n^{\pm1}]$ be a Laurent polynomial.
Then, we have

\[
\]

Let $\mathcal{F}=\bigoplus_{i,j} \alpha_{ij} x^i y^j$ be a tropical polynomial.
We write $\Newt(\mathcal{F})\subset \mathbb{R}^2$ for the convex hull of the set $\{(i, j)\in \mathbb{Z}^2\ |\ \alpha_{ij}\neq -\infty\}$.
Let $A_{\mathcal{F}}\subset \mathbb{R}^3$ be the convex hull of the set

\[
\{ (i, j, \alpha)\in \mathbb{Z}^2\times \mathbb{R}\ \mid \alpha \leq \alpha_{ij}\}.
\]

Then, the projections of the bounded faces of $A_{\mathcal{F}}$ form a lattice subdivision of $\Newt(\mathcal{F})$.
This naturally has a structure of a polyhedral complex.
The dual subdivision of $\mathcal{F}$ is this polyhedral complex and we denote it by $\Delta_{\mathcal{F}}$.
For a Laurent polynomial $f\in k[x^{\pm1}, y^{\pm1}]$, we also write $\Delta_f$ for the dual subdivision of $\trop(f)$.

Let $\Gamma=V(\mathcal{F})$ be a tropical plane curve.
Then, $\Gamma$ is the support of a $1$-dimensional polyhedral complex $\Sigma_{\mathcal{F}}$ (possibly with noncompact support) in $\mathbb{R}^2$.
It is dual to the subdivision $\Delta_{\mathcal{F}}$ in the following sense:

* (Closures of) domains of $\mathbb{R}^2\setminus \Gamma$ correspond to lattice points in $\Delta_\mathcal{F}$.

* $1$-dimensional cells in $\Sigma_{\mathcal{F}}$ correspond to $1$-simplexes in $\Delta_{\mathcal{F}}$.

* $0$-dimensional cells in $\Sigma_{\mathcal{F}}$ correspond to $2$-dimensional cells in $\Delta_{\mathcal{F}}$.

* This correspondence is inclusion-reversing.

* A $1$-dimensional cell in $\Sigma_{\mathcal{F}}$ is orthogonal to the corresponding $1$-simplex in $\Delta_{\mathcal{F}}$ (see Figure <ref>).

In particular, $1$-dimensional cells in $\Sigma_{\mathcal{F}}$ have rational slopes.

(L1) at (2,0);
(L2) at (2.5,0);
(L3) at (3,0);
(L4) at (2,0.5);
(L5) at (2.5,0.5);
(L6) at (2,1);

(L7) at (-1,1);
(L8) at (-0.5,1);
(L9) at (0,0.5);
(L10) at (0.5,0.5);
(L11) at (1,1);
(L12) at (0,0);
(L13) at (0.5,0);
(L14) at (-1,1.5);
(L15) at (-0.5,1.5);
(L16) at (0,2);

(A1) at (7,0);
(A2) at (7.5,0);
(A3) at (8,0);
(A4) at (7,0.5);
(A5) at (7.5,0.5);
(A6) at (7,1);

(A7) at (4,0.5);
(A8) at (4.5,0.5);
(A9) at (5,1);
(A10) at (5.5,1);
(A11) at (6,1.5);
(A12) at (4.5,0);
(A13) at (5.5,0);
(A14) at (4,1.5);
(A15) at (5,1.5);
(A16) at (5.5,2);

[label=below:$\Delta_{\mathcal{F}}$] (a) at (2.5,-0.25);
[label=below:$\Delta_{\mathcal{G}}$] (b) at (7.5,-0.25);
[label=below:$V(\mathcal{F})$] (a) at (0,-0.25);
[label=below:$V(\mathcal{G})$] (b) at (5,-0.25);

[label=below:$\leftrightarrow$] (a) at (1.5,0.7);
[label=below:$\leftrightarrow$] (b) at (6.5,0.7);

(Smooth) tropical plane curves and their dual subdivisions.

Let $\Gamma$ be a tropical plane curve.
We call a $0$-dimensional cell of $\Gamma$ a vertex of $\Gamma$ and a $1$-dimensional cell of $\Gamma$ an edge of $\Gamma$.

We define the multiplicity of an edge of a tropical plane curve using the dual subdivision.

Let $\Gamma=V(\mathcal{F})$ be a tropical plane curve and $\sigma \in \Sone(\Gamma)$ an edge of $\Gamma$.
The multiplicity $m_{\sigma}$ of $\sigma$ in $\Gamma$ is the lattice length of the corresponding $1$-simplex of $\Delta_{\mathcal{F}}$.

From now on, a tropical curve will refer to the polyhedral set $\Gamma$ together with multiplicities on its edges.

Let $\mathcal{F}=\bigoplus_{i,j} \alpha_{ij} x^i y^j$ be a tropical polynomial.
Let $\sigma$ be an edge of $\Sigma_{\mathcal{F}}$ with multiplicity $1$ and $\sembunioil$ the corresponding $1$-simplex of $\Delta_{\mathcal{F}}$.
Assume that $\alpha_{ij}+ip+jq=\mathcal{F}(p, q)$ for $(p, q)\in \sigma$ and $(i, j)\in \mathbb{Z}^2\setminus \{\bio, \bil\}$.
Then $(i, j)$ is one of the vertices of a $2$-dimensional cell in $\Delta_{\cF}$ containing $\sembunioil$ as its face.
In particular, for any $(i, j)\in (\Aff(\sembunioil)\cap \mathbb{Z}^2)\setminus \{\bio, \bil\}$, we have

\[
\alpha_{ij}+ip+jq<\mathcal{F}(p, q).
\]

Let $\Gamma$ be a tropical plane curve, $P$ a vertex of $\Gamma$ and $L$ an edge of $\Gamma$ containing $P$.
Let $R$ be the ray which contains $L$ such that $P$ is its endpoint.
We denote by $\mathbf{v}_{P, L}$ the primitive vector that have the same direction as $R$.

Tropical plane curves satisfy the following balancing condition.

Let $\Gamma$ be a tropical plane curve and $P$ a vertex of $\Gamma$ and $L_1, \dots, L_n$ the edges of $\Gamma$ containing $P$ with multiplicities $m_{L_i}$.
Then, we have

\[
\sum_{i}m_{L_i}\mathbf{v}_{P, L_i}=\mathbf{0}.
\]

Let $\Gamma=V(\mathcal{F})$ be a tropical plane curve.
A vertex $P\in \Gamma$ is called smooth if the area of the corresponding cell in $\Delta_{\mathcal{F}}$ is $1/2$.
We see that this is equivalent to the condition that it is trivalent and all the multiplicities of the three edges $L_1$, $L_2$ and $L_3$ containing $P$ are $1$, and for some (or any) pair $(i, j)$ ($i, j\in \{1, 2, 3\}$, $i\neq j$), we have $|\det(\mathbf{v}_{P, L_i}, \mathbf{v}_{P, L_j})|=1$.

A tropical plane curve $\Gamma=V(\mathcal{F})$ is called smooth if all the lattice lengths of the $1$-simplexes of $\Delta_{\mathcal{F}}$ are $1$ and all the areas of the $2$-dimensional cells of $\Delta_{\mathcal{F}}$ are $1/2$ (see Figure <ref>).
In other words, $\Gamma$ is smooth if all the vertices are smooth and all the multiplicities of the edges are $1$.

Let $\Gamma$ be a tropical plane curve and $\sigma$ an edge of $\Gamma$.
We denote by $\mathbf{v}_{\sigma}$ primitive vector that have the same direction as $\Aff(\sigma)$.
This is well-defined up to sign.

Let $\Gamma_1$ and $\Gamma_2$ be tropical plane curves.
A point $P$ is a transverse intersection point of $\Gamma_1$ and $\Gamma_2$ if it is a proper intersection point of them and is not a vertex of either of them.
For a transverse intersection point $P$, there exist unique edges $L_i\in \Sone(\Gamma_i)$ ($i=1, 2$) containing $P$ in their interiors.
In this case, we say that $L_1$ and $L_2$ intersect transversely at $P$, and we define the intersection multiplicity at $P$ as

\[
i(P; \Gamma_1 \cdot \Gamma_2):=m_{L_1}m_{L_2}|\det(\mathbf{v}_{L_1}, \mathbf{v}_{L_2})|.
\]

Tropical plane curves $\Gamma_1$ and $\Gamma_2$ intersect transversely if all the points in $\Gamma_1\cap \Gamma_2$ are transverse intersection points.

Let $\Gamma_1$ and $\Gamma_2$ be tropical plane curves intersecting transversely at a point $P$.
Assume that $P$ is contained in edges $L_i\in \Sone(\Gamma_i)$ ($i=1, 2$) of $\Gamma_i$.
Let $\overline{(a, b)(c, d)}$ be the corresponding cell in $\Delta_{\Gamma_1}$ of $L_1$ and $\overline{(e, f)(g, h)}$ the corresponding cell in $\Delta_{\Gamma_2}$ of $L_2$.
Then, we have

\[
i(P; \Gamma_1 \cdot \Gamma_2)
c-a & g-e \\
d-b & h-f \\
\end{pmatrix}\right|.
\]

For tropical plane curves $\Gamma_1$ and $\Gamma_2$, it is known that for a generic vector $\mathbf{v}\in \mathbb{R}^2$ and a real number $\epsilon \in \mathbb{R}$ with sufficiently small absolute value, $\Gamma_1+\epsilon \mathbf{v}$ and $\Gamma_2$ intersect transversely, and the following sum is well-defined (see <cit.>).

Let $\Gamma_1$ and $\Gamma_2$ be tropical plane curves.
We define the intersection multiplicity at a point $P\in \Gamma_1 \cap \Gamma_2$ as

\[
i(P; \Gamma_1 \cdot \Gamma_2):=\sum_{L_1 \ni P, L_2\ni P}\left(\sum_{Q\in (L_1+\epsilon \mathbf{v})\cap L_2}i((Q; (\Gamma_1+\epsilon \mathbf{v}) \cdot \Gamma_2)\right),
\]

where $\mathbf{v}\in \mathbb{R}^2$ is a generic vector, $\epsilon \in \mathbb{R}$ is a sufficiently small real number.

Note that we have $i(P; \Gamma_1 \cdot \Gamma_2)=i(P; \Gamma_2 \cdot \Gamma_1)$.
It was shown in <cit.> that the following limit exists and is independent of a generic vector $\mathbf{v}\in \mathbb{R}^2$, allowing us to define the stable intersections.

Let $\Gamma_1$ and $\Gamma_2$ be tropical plane curves.
Let $\mathbf{v}\in \mathbb{R}^2$ be a generic vector and $\epsilon \in \mathbb{R}$ a real number.
Note that $\Gamma_1+\epsilon \mathbf{v}$ is also a tropical curve, whose dual subdivision is the same as that of $\Gamma_1$.
The stable intersection of $\Gamma_1$ and $\Gamma_2$ is the limit

\[
\lim_{\epsilon \to 0}((\Gamma_1+\epsilon \mathbf{v})\cap\Gamma_2).
\]

Let $\Gamma_1$ and $\Gamma_2$ be tropical plane curves and $S$ the stable intersection of $\Gamma_1$ and $\Gamma_2$.
The stable intersection divisor of $\Gamma_1$ and $\Gamma_2$ is defined as

\[
\sum_{P\in S}i(P; \Gamma_1 \cdot \Gamma_2)P.
\]

We cite a couple of theorems concerning the tropicalizations of intersections of hypersurfaces when their tropicalizations do not necessarily intersect properly.
Note that we will use these theorems for tropical plane curves.

Let $X_1, X_2\subset (k^*)^n$ be subvarieties.
Then, there exists a Zariski dense subset $U\subset (k^*)^n$ such that

\[
U\subset \{\mathbf{t}\in (k^*)^n\ |\ \val(\mathbf{t})=\mathbf{0}\}
\]

\[
\trop(X_1\cap \mathbf{t}X_2)=\Trop(X_1)\cdot \Trop(X_2)\ \text{for all $\mathbf{t}\in U$},
\]

where the right hand side is the stable intersection.

For the scheme theoretic version of this theorem, see <cit.>.

For a scheme $X$, let $|X|$ denote the set of closed points of $X$.

The following theorem says that the tropicalization conserves the intersection number in a certain sense.

Let $X_1, \dots, X_m\in (k^*)^n$ be pure dimensional closed subschemes of $(k^*)^n$ with $\sum_{i}\codim (X_i)=n$.
Let $C$ be a connected component of $\bigcap_{i}\Trop(X_i)$, and suppose that $C$ is bounded.
Then there are only finitely many points $x\in |\bigcap_{i}X_i|$ with $\trop(x)\in C$, and

\[
\sum_{\substack{x\in |\bigcap_{i}X_i|\\\trop(x)\in C}}i_k(x; X_1\cdots X_m)=\sum_{P\in C}i(P; \Trop(X_1)\cdots \Trop(X_m)).
\]

§ PREPARATIONS FOR THE MAIN THEOREMS

We will provide additional explanations for some facts from Section 1 and make preparations for the next section.
Since we are going to compare the valuations of different terms in polynomials, we make the following definition.

We define a map $\tau$ as follows:

\begin{eqnarray*}
\tau: \mathbb{T}[x^{\pm1}, y^{\pm1}]\times \mathbb{Z}^2\times \mathbb{R}^2&\to&\mathbb{R}\cup \{-\infty\}\\
(\bigoplus_{\mathbf{i}} \alpha_{\mathbf{i}} \mathbf{x}^{\mathbf{i}}; \mathbf{j}; P)&\mapsto&\alpha_{\mathbf{j}}+\mathbf{j}\cdot P.
\end{eqnarray*}

For a Laurent polynomial $f\in k[x^{\pm1}, y^{\pm1}]$, we write $\tau(f; \mathbf{j}; P)$ for $\tau(\trop(f); \mathbf{j}; P)$.

Let $\cF$ be a bivariate tropical polynomial and $\mathbf{i}_0, \mathbf{i}_1\in \mathbb{Z}^2$ ($\mathbf{i}_0\neq \mathbf{i}_1$) two distinct lattice points.
Let $P\in \mathbb{R}^2$ be a point and $l=\Aff(\sembunioil)$.
We define $\tau(\cF; \sembunioil; P)$ and $\tau'(\cF; \sembunioil; P)$ by

\begin{eqnarray*}
\tau(\cF; \sembunioil; P)&:=&\max \{ \tau(\cF; \mathbf{i}; P)\ |\ \mathbf{i}\in l\cap \mathbb{Z}^2 \},\\
\tau'(\cF; \sembunioil; P)&:=&\max \{ \tau(\cF; \mathbf{i}; P)\ |\ \mathbf{i}\in (l\cap \mathbb{Z}^2)\setminus\{\mathbf{i}_0, \mathbf{i}_1\} \}.
\end{eqnarray*}

For $f\in k[x^{\pm1}, y^{\pm1}]$, we write $\tau(f; \sembunioil; P)$ and $\tau'(f; \sembunioil; P)$ for $\tau(\trop(f); \sembunioil; P)$ and $\tau'(\trop(f); \sembunioil; P)$, respectively.

The map $\tau$ satisfies the following.

Let $f, g\in k[x^{\pm1}, y^{\pm1}]$ be Laurent polynomials.
Then, for all $\mathbf{i}\in \mathbb{Z}^2$ and $P\in \mathbb{R}^2$, we have

\[
\tau(f+g; \mathbf{i}; P)\leq \max \{ \tau(f; \mathbf{i}; P), \tau(g; \mathbf{i}; P)\}.
\]

Moreover, the equality holds if $\tau(f; \mathbf{i}; P)\neq \tau(g; \mathbf{i}; P)$.

This is clear from the ultrametric inequality for the valuation.

Recall that $\cLSFG$ is the set of line segments $L$ satisfying the following:

* $L$ is a connected component of the intersection $V(\cF)\cap V(\cG)$.

* The intersection multiplicity of $V(\cF)$ and $V(\cG)$ on $L$ is  $2$.

Let $\cF$ and $\cG$ be bivariate tropical polynomials.

* Let $L\in \cLSFG$, and $P_+$ and $P_-$ the endpoints of $L$.
Then, $P_*$ ($P_*=P_+, P_-$) is a smooth vertex in one of $V(\cF)$ and $V(\cG)$, and is in the interior of an edge of multiplicity $1$ in the other.
Furthermore, $V(\cF)$ and $V(\cG)$ do not contain vertices in the interior of $L$.
In other words, we have the following:
For a neighborhood $U$ of $L$, the restrictions of $V(\cF)$ and $V(\cG)$ to $U$ are as in Figure <ref>, where each vertex is smooth and each edge has multiplicity $1$.

In particular, the stable intersection points of $V(\cF)$ and $V(\cG)$ on $L$ are the endpoints of $L$, each with multiplicity $1$.

* For $L\in \cRayFG$, the endpoint of $L$ is a smooth vertex in one of $V(\cF)$ and $V(\cG)$, and is in the interior of an edge of multiplicity $1$ in the other.
Furthermore, $V(\cF)$ and $V(\cG)$ do not contain vertices in the interior of $L$.

[label=left:$P_-$] (A1) at (0.4,0.4);
[label=left:$P_+$] (A2) at (0.4,1.6);
[label=right:$P_-$] (A3) at (1.6,0.4);
[label=right:$P_+$] (A4) at (1.6,1.6);
[label=left:$P_-$] (A5) at (3.4,0.4);
[label=left:$P_+$] (A6) at (3.4,1.6);
[label=right:$P_-$] (A7) at (4.6,0.4);
[label=right:$P_+$] (A8) at (4.6,1.6);
[label=left:$P_-$] (A9) at (6.4,0.4);
[label=left:$P_+$] (A10) at (6.4,1.6);
[label=right:$P_-$] (A11) at (7.6,0.4);
[label=right:$P_+$] (A12) at (7.6,1.6);
[label=left:$P_-$] (A13) at (9.4,0.4);
[label=left:$P_+$] (A14) at (9.4,1.6);
[label=right:$P_-$] (A15) at (10.6,0.4);
[label=right:$P_+$] (A16) at (10.6,1.6);

[dotted] (0,0)–(0.8,0)–(0.8,2)–(0,2)–cycle;
[dotted] (1.2,0)–(2,0)–(2,2)–(1.2,2)–cycle;

[dotted] (3,0)–(3.8,0)–(3.8,2)–(3,2)–cycle;
[dotted] (4.2,0)–(5,0)–(5,2)–(4.2,2)–cycle;

[dotted] (6,0)–(6.8,0)–(6.8,2)–(6,2)–cycle;
[dotted] (7.2,0)–(8,0)–(8,2)–(7.2,2)–cycle;

[dotted] (9,0)–(9.8,0)–(9.8,2)–(9,2)–cycle;
[dotted] (10.2,0)–(11,0)–(11,2)–(10.2,2)–cycle;

in 1,...,16 [black] (A) circle (0.045);

[label=below:(a)] (a) at (1,-0.1);
[label=below:(b)] (b) at (4,-0.1);
[label=below:(c)] (c) at (7,-0.1);
[label=below:(d)] (d) at (10,-0.1);

$V(\cF)$ and $V(\cG)$ in a neighborhood of $L\in \cLSFG$.

First, note that each vertex of $L$ is a vertex of at least one of $V(\cF)$ and $V(\cG)$, and that if $P\in L$ is a vertex of $V(\cF)$ or $V(\cG)$, then we have $i(P; V(\cF)\cdot V(\cG))\geq 1$.
We show the statement on $L\in \cLSFG$.
The case of $L\in \cRayFG$ is similar.
We can assume that $L$ is contained in the $y$-axis and $P_+=(0, a_1)$ and $P_-=(0, a_2)$ with $a_1>a_2$ by applying an affine transformation with an unimodular integral coefficient matrix.
Let $\epsilon>0$ be a sufficiently small positive number and $\mathbf{v}=(v_1, v_2)\in \mathbb{R}^2$ a general vector such that $v_1, v_2>0$ and $v_2/v_1$ is sufficiently large.
We will consider $V(\cF)\cap (\epsilon \mathbf{v_i}+V(\cG))$ and $V(\cG)\cap (\epsilon \mathbf{v_i}+V(\cF))$.
Let $U_1$ be a sufficiently small neighborhood of $P_+$ and $D_R:=\{(p_1, p_2)\in \mathbb{R}^2\ |\ p_1>0\}$.
Let $L_1, \dots, L_s$ be the edges of $V(\cF)$ which intersect $U_1$, and $L'_1, \dots, L'_t$ the edges of $V(\cG)$ which intersect $U_1$, with $L_1\supset L$ and $L'_1\supset L$.

Assume that $P_+$ is a vertex in both of $V(\cF)$ and $V(\cG)$, i.e. $s\geq 3$ and $t\geq 3$.
Then, by the balancing condition at $P_+$, there are $i$ and $j$ such that

\[
D_R\cap L_i\neq \emptyset \text{ and }D_R\cap L'_j\neq \emptyset.
\]

Note that by the assumption that $L\in \cLSfg$ is a connected component of $V(\cF) \cap V(\cG)$, we have $L_i\cap L'_j=\{P_+\}$.

Thus, we have $L_i\cap (\epsilon \mathbf{v}+L'_j)\neq \emptyset$ and $L_i\cap (\epsilon \mathbf{v}+L'_1)\neq \emptyset$, or $L'_j\cap (\epsilon \mathbf{v}+L_i)\neq \emptyset$ and $L'_j\cap (\epsilon \mathbf{v}+L_1)\neq \emptyset$.
Hence, we have $i(P_+; V(\cF) \cdot V(\cG))=i(P_+; V(\cG) \cdot V(\cF))\geq 2$.
However, we can see that $P_-$ is also a stable intersection point, and hence the intersection multiplicity on $L$ would be greater than $3$.
This contradicts the assumption.
Thus $P_+$ is a vertex of exactly one of $V(\cF)$ and $V(\cG)$.
Assume that $P_+$ is a vertex of $V(\cF)$.
Since the intersection multiplicity at $P_1$ is $1$, it is cleat that exactly one of $\{L_2, \dots, L_s\}$, say $L_2$, intersects $D_R$ and its multiplicity and that of $L'_1$ are both $1$ and that $|\det(\mathbf{v}_{L_2}, \mathbf{v}_{L'_1})|=1$.
By the balancing condition, $P_+$ is a smooth vertex of $V(\cF)$.
The same holds at the point $P_-$.
Thus, the intersection multiplicity of $V(\cF)$ and $V(\cG)$ on the endpoints of $L$ is 2, and hence, $V(\cF)$ and $V(\cG)$ do not contain vertices in the interior of $L$.

Let $\cF=\bigoplus_{\mathbf{i}}\alpha_{\mathbf{i}}\mathbf{x}^{\mathbf{i}}, \cG=\bigoplus_{\mathbf{i}}\beta_{\mathbf{i}}\mathbf{x}^{\mathbf{i}}\in \mathbb{T}[x^{\pm1}, y^{\pm1}]$ be tropical polynomials, $L\in \cSI(\cF, \cG)$, and $P_+$ an endpoint of $L$.
Assume that $\Phi_1(L)=\Phi_2(L)=\sembunioil$, $\alpha_{\bio}=\beta_{\bio}$ and $\alpha_{\bil}=\beta_{\bil}$.
Let $U$ be a sufficiently small neighborhood of $P_+$ and $\overline{L}=\Aff(L)$.
By Lemma <ref>, for either $(\cF_1, \cF_2)=(\cF, \cG)$ or $(\cF_1, \cF_2)=(\cG, \cF)$, the point $P_+$ is a smooth vertex of $V(\cF_1)$ and is in the interior of an edge of multiplicity $1$ in $V(\cF_2)$.
Let $\sigma_+$ be the $2$-simplex of $\Delta_{\cF_1}$ corresponding to $P_+$ and $\bip$ the vertex of $\sigma_+$ other than $\bio$ and $\bil$.
Then, for all $P\in U\cap (\overline{L}\setminus L)$, we have

\begin{eqnarray*}
\tau(\cF_1; \bip; P)>\tau(\cF; \mathbf{i}; P), \tau(\cG; \mathbf{i}; P), \tau(\cF_2; \bip; P)\ (\mathbf{i}\in \mathbb{Z}^2\setminus \{\bip\}),
\end{eqnarray*}

\begin{eqnarray*}
\tau(\cF_1; \bip; P_+)&=&\tau(\cF; \bi; P_+)=\tau(\cG; \bi; P_+)\ (\text{$\bi=\bio$ or $\bil$})\\
&>&\tau(\cF; \mathbf{j}; P_+), \tau(\cG; \mathbf{j}; P_+), \tau(\cF_2; \bip; P_+)\ (\mathbf{j}\in \mathbb{Z}^2\setminus \{\bio, \bil, \bip\}).
\end{eqnarray*}

By symmetry, we may assume that $\cF_1=\cF$.

Let $P\in  U\cap (\overline{L}\setminus L)$.
Then, the restrictions of the two tropical plane curves are as in Figure <ref>.
We have

\begin{eqnarray}
\tau(\cF; \bip; P)>\tau(\cF; \mathbf{i}; P)\ \ (\mathbf{i}\in \mathbb{Z}^2\setminus \{\bip\}),
\end{eqnarray}

and for all $\mathbf{j}\in \mathbb{Z}^2\setminus \{\bio, \bil, \bip\}$, we have

\begin{eqnarray}
\tau(\cF; \bip; P_+)=\tau(\cF; \bio; P_+)=\tau(\cF; \bil; P_+)>\tau(\cF; \mathbf{j}; P_+).
\end{eqnarray}

For all $\mathbf{i}\in \mathbb{Z}^2\setminus \{\bio, \bil\}$, we have

\begin{eqnarray}
&\tau(\cG; \bio; P)=\tau(\cG; \bil; P)>\tau(\cG; \mathbf{i}; P),&\\
&\tau(\cG; \bio; P_+)=\tau(\cG; \bil; P_+)>\tau(\cG; \mathbf{i}; P_+).&
\end{eqnarray}

By the assumption that $\alpha_{\bio}=\beta_{\bio}$ and $\alpha_{\bil}=\beta_{\bil}$, we have

\begin{eqnarray}
\ \tau(\cF; \bi; P)=\tau(\cG; \bi; P),\ \tau(\cF; \bi; P_+)=\tau(\cG; \bi; P_+) \ \ \text{($\bi=\bio$ or $\bil$)}.
\end{eqnarray}

By the inequalities (1), (3) and (5), we have

\begin{eqnarray*}
\tau(\cF; \bip; P)>\tau(\cF; \mathbf{i}; P), \tau(\cG; \mathbf{i}; P), \tau(\cG; \bip; P)\ \ (\mathbf{i}\in \mathbb{Z}^2\setminus \{\bip\}),
\end{eqnarray*}

and by the inequalities (2), (4) and (5), we have

\begin{eqnarray*}
\tau(\cF; \bip; P_+)&=&\tau(\cF; \bi; P_+)=\tau(\cG; \bi; P_+)\ \ (\text{$\bi=\bio$ or $\bil$})\\
&>&\tau(\cF; \mathbf{j}; P_+), \tau(\cG; \mathbf{j}; P_+), \tau(\cG; \bip; P_+)\ \ (\mathbf{j}\in \mathbb{Z}^2\setminus \{\bio, \bil, \bip\}).
\end{eqnarray*}

(0,0) circle [radius=1.2];
(4,0) circle [radius=1.2];
(-0.7,-0.45) circle [radius=0.275];
(0.7,-0.45) circle [radius=0.275];
(0,0.8) circle [radius=0.275];
(3.3,0) circle [radius=0.275];
(4.8,0) circle [radius=0.275];

(A1) at (0,0);
[label=right:$P_+$] (B1) at (-0.05,-0.1);
(A2) at (0,0.3);
[label=right:$P$] (D1) at (-0.05,0.325);
(A3) at (0,-2);
(A4) at (4,0);
[label=right:$P_+$] (C1) at (3.95,-0.1);
(A5) at (4,0.3);
[label=right:$P$] (C2) at (3.95,0.33);
(A6) at (4,-2);
[label=right:$L$] (A7) at (0,-1.6);
[label=right:$L$] (A8) at (4,-1.6);
[label=above:$\overline{L}$] (B2) at (0,1.4);
[label=above:$\overline{L}$] (B3) at (4,1.4);
[label=below:$U$] (B4) at (-0.8,1.4);
[label=below:$U$] (B5) at (3.2,1.4);

[label=above:$\bip$] (a1) at (0.01,0.53);
[label=above:$\bio$] (a2) at (-0.7,-0.69);
[label=above:$\bil$] (a3) at (0.7,-0.69);
[label=above:$\bio$] (a4) at (3.3,-0.24);
[label=above:$\bil$] (a5) at (4.8,-0.24);

[very thick] (A1)–(A3);
[very thick] (A4)–(A6);
[dotted, thick] (A1)–(0,0.525);
[dotted, thick] (0,1.075)–(0,1.4);
[dotted, thick] (A3)–(0,-2.3);
[dotted, thick] (A6)–(4,-2.3);

in 1,2,4,5 [black] (A) circle (0.06);

[label=below:$V(\cF_1)$] (a) at (0,-2.3);
[label=below:$V(\cF_2)$] (b) at (4,-2.3);

$V(\cF_1)$ and $V(\cF_2)$ in a neighborhood $U$ of $P_+$.

In the following, we show some lemmas.
We will consider a Laurent polynomial of the form $G=g+h(\mathbf{x}^{\mathbf{v}})f$, where $h\in k[t^{\pm1}]$ is a univariate Laurent polynomial.

Given a Laurent polynomial $F=\sum_i\alpha_ix^i\in k[x^{\pm1}]$, we define

\begin{eqnarray*}
v'(F)&=&\min\{\val(\alpha_i)\ |\ i\neq 0, 1\}.
\end{eqnarray*}

Now we are going to find an element of the ideal $(f, g)$ that is useful in studying $\trop(V(f)\cap V(g))$.
The proof of the following lemma gives an algorithm to find this element.

Let $\de \geq 0$ be a non-negative number.
Let $f=\sum_{i, j}c_{ij}x^iy^j, g=\sum_{i, j}d_{ij}x^iy^j\in k[x^{\pm1}, y^{\pm1}]$ be Laurent polynomials satisfying the following:

* $\val(c_{\bio})=\val(d_{\bio})\neq \infty$,  $\val(c_{\bil})=\val(d_{\bil})\neq \infty$.

* $\bil-\bio$ is primitive.

* $\mu(f; \sembunioil), \mu(g; \sembunioil)>0$.

Then, there exists a Laurent polynomial $h(t)=\sum_{i}a_it^i\in k[t^{\pm1}]$ satisfying the following conditions:

* For all $i\in \mathbb{Z}$, we have $\val(a_i)>i\val\left(\frac{c_{\mathbf{i}_1}}{c_{\mathbf{i}_0}}\right)$.

* For the Laurent polynomial $g':=g+h(\mathbf{x}^{\mathbf{j}-\mathbf{i}})f=\sum_{i, j}d'_{ij}x^iy^j$, we have

\begin{eqnarray*}
&\val(d'_{\mathbf{i}_0})=\val(d_{\mathbf{i}_0}),& \\
&\val(d'_{\mathbf{i}_1})=\val(d_{\mathbf{i}_1}),& \\
&\mu(g'; \sembunioil)>\de.&
%&\forall n\in \mathbb{Z}\setminus \{0, 1\}, \val\left( \frac{d'_{\mathbf{i}_0+n(\mathbf{i}_1-\mathbf{i}_0)}}{d'_{\mathbf{i}_0}}\right)-n\val\left(\frac{d'_{\mathbf{i}_1}}{d'_{\mathbf{i}_0}}\right)>\de. &
\end{eqnarray*}

We can assume that $\mathbf{i}_0=(0, 0)$ and $\mathbf{i}_1=(1, 0)$ by applying an affine transformation.
Then, the statements are concerned only about the coefficients of $x^i$, and we can assume that $f=\sum_{i}c_ix^i, g=\sum_{i}d_ix^i\in k[x^{\pm1}]$.
We can also assume that $c_0=1$ by multiplying a unit.
By changing the variable $x$ to $c_1x$, we may also assume $c_1=1$.
Then, we have $v'(f)>0$ and $v'(g)>0$.
It is sufficient to show that there exists a Laurent polynomial $h=\sum_{i}a_ix^i\in k[x^{\pm1}]$ with $v(h)>0$ such that for the Laurent polynomial $g':=g+hf$, we have $v_0(g')=v_1(g')=0$ and $v'(g')>\de$.
Let $\de_0=v'(f)$ and $\de_1=v'(g)$.
Given a Laurent polynomial $F\in k[x^{\pm1}]$, we define

\begin{eqnarray*}
\Mm(F)&=&\min\{n\in \mathbb{Z}\ |\ n\leq0,\ v_n(F)<\de_0+\de_1\},\\
\Mp(F)&=&\max\{n\in \mathbb{Z}\ |\ n\geq1,\ v_n(F)<\de_0+\de_1\}.
\end{eqnarray*}

Note that $\Mm(F)\leq 0$, $\Mp(F)\geq 1$ and that for Laurent polynomials $F_1, F_2\in k[x^{\pm1}]$, we have

\begin{eqnarray*}
\Mm(F_1+F_2)&\geq& \min\{\Mm(F_1), \Mm(F_2)\},\\
\Mp(F_1+F_2)&\leq& \max\{\Mp(F_1), \Mp(F_2)\}.
\end{eqnarray*}

If $\Mp(g)-\Mm(g)>1$, then there exist $a\in k$ and $i\in \mathbb{Z}$ such that $\val(a)>0$ and

\begin{eqnarray*}
\Mp(g-ax^if)-\Mm(g-ax^if)<\Mp(g)-\Mm(g).
\end{eqnarray*}

Assume that $n:=\Mp(g)>1$.
Let $a=d_n$.
Then, we have $\val(a)\geq \de_1 (>0)$.
Let $-ax^{n-1}f=\sum_{i}\alpha_ix^i$ and $g-ax^{n-1}f=\sum_{i}\beta_ix^i$.
Then, we have $\val(\beta_n)=\val(0)=\infty$ and

\begin{eqnarray*}
i< n-1 &\Rightarrow& \val(\alpha_i)\geq\de_0+\de_1,\\
i=n-1, n &\Rightarrow& \val(\alpha_i)=\val(a)\geq \de_1,\\
n<i &\Rightarrow& \val(\alpha_i)\geq\de_0+\de_1.
\end{eqnarray*}

Thus, we have

\begin{eqnarray*}
n\leq i \Rightarrow \val(\beta_i)\geq\de_0+\de_1,
\end{eqnarray*}

and $\Mm(-ax^{n-1}f)=0$.
Hence, we have

\begin{eqnarray*}
\Mp(g-ax^{n-1}f)<n=\Mp(g),
\end{eqnarray*}

\begin{eqnarray*}
\Mm(g-ax^{n-1}f)\geq \min\{\Mm(g), \Mm(-ax^{n-1}f)\}\geq \Mm(g).
\end{eqnarray*}

Therefore, we have

\begin{eqnarray*}
\Mp(g-ax^{n-1}f)-\Mm(g-ax^{n-1}f)<\Mp(g)-\Mm(g).
\end{eqnarray*}

The case where $\Mm(f)<0$ can be shown similarly.

Note that in the proof of the above claim, we have $\val(\beta_0)=\val(\beta_1)=0$.
From the above claim, we can show by induction on $n=\Mp(g)-\Mm(g)$ that there exists a Laurent polynomial $h_0=\sum_{i}\alpha_ix^i\in k[x^{\pm1}]$ with $v(h_0)>0$ such that for the Laurent polynomial $g_1:=g+h_0f=\sum_{i}\beta_ix^i$, we have $v_0(g_1)=v_1(g_1)=0$ and $v'(g_1)>\de_0+\de_1$.
Then, by induction on $\max\{0, \left[\frac{\de-v'(g)}{v'(f)}\right]\}$, we may ensure that there exists $h\in k[x^{\pm1}]$ with $v(h)>0$ such that for $g':=g+hf$, we have $v_0(g')=v_1(g')=0$ and $v'(g')>\de$.

Let $\de \geq 0$ be a non-negative number and $f=\sum_{i, j}c_{ij}x^iy^j, g=\sum_{i, j}d_{ij}x^iy^j\in k[x^{\pm1}, y^{\pm1}]$ Laurent polynomials satisfying the assumption of Lemma <ref>.
In this case, we define $h(\de; g; f; \sembunij)\in k[t^{\pm1}]$ by the Laurent polynomial $h\in k[t^{\pm1}]$ obtained by the algorithm in the proof of Lemma <ref>.
We also define $G(\de; g; f; \sembunij)\in k[x^{\pm1}, y^{\pm1}]$ by

\begin{eqnarray*}
G(\de; g; f; \sembunij):=g+h(\de; g; f; \sembunij)(\mathbf{x}^{\mathbf{j}-\mathbf{i}})f-\frac{d'_{\bj}}{c'_{\bj}}\Bigl(f+h(\de; f; f; \sembunij)(\mathbf{x}^{\mathbf{j}-\mathbf{i}})f\Bigr),
\end{eqnarray*}

\begin{eqnarray*}
g+h(\de; g; f; \sembunij)(\mathbf{x}^{\mathbf{j}-\mathbf{i}})f:=\sum_{i, j}d'_{ij}x^iy^j,\\
f+h(\de; f; f; \sembunij)(\mathbf{x}^{\mathbf{j}-\mathbf{i}})f:=\sum_{i, j}c'_{ij}x^iy^j.
\end{eqnarray*}

We write $\mathbf{e}_1=(1, 0), \mathbf{e}_2=(0, 1)\in \mathbb{R}^2$ for the standard basis.

To compare the tropicalizations of $V(f)$, $V(g)$ and $V(G(\de; g; f; \sembunij))$, we use the following lemma.

Let $f=\sum_{i, j}c_{ij}x^iy^j\in k[x^{\pm1}, y^{\pm1}]$ be a Laurent polynomial such that the dual subdivision $\Delta_f$ contains a $1$-simplex $\sigma=\sembunioil$, where $\mathbf{v}:=\bil-\bio$ is primitive.
Let $L$ be the edge of $\Trop(V(f))$ corresponding to $\sigma$ and $P\in L$.
Let $h(t)=\sum_{i}a_it^i \in k[t^{\pm1}]$ be a Laurent polynomial that satisfies

\[
\val(a_i)>i\val\left(\frac{c_{\bil}}{c_{\bio}}\right)
\]

for all $i\in \mathbb{Z}$.
Then, for any $\mathbf{i}\in \mathbb{Z}^2$, we have

\[
\tau(h(\mathbf{x}^\mathbf{v})f; \mathbf{i}; P)< \tau(f; \overline{\bi,\bi+\mathbf{v}}; P).
\]

We can assume that $\bio=(0, 0)$ and $\bil=(1, 0)$ by applying an affine transformation.
We can assume that $\val(c_{00})=0$ by multiplying a unit to $f$.
Let $P\in L$ and $\mathbf{i}\in \mathbb{Z}^2$.
Since $L$ is the edge of  $\Trop(V(f))$ corresponding to $\sigma=\sembunoolo$, we have

\[
0=-\val(c_{00})=\tau(f; (0, 0); P)=\tau(f; (1, 0); P)=-\val(c_{10})+\mathbf{e}_1\cdot P,
\]

and hence,

\[
\val(c_{10})=\mathbf{e}_1\cdot P.
\]

We have

\[
h(\mathbf{x}^{\mathbf{e}_1})f(\mathbf{x})=\sum_{s}\left(\sum_{\mathbf{i}}\left(\sum a_{s}c_{\mathbf{i}-s\mathbf{e}_1}\right)\mathbf{x}^{\mathbf{i}}\right),
\]

and therefore

\[
\tau(h(\mathbf{x}^{\mathbf{e}_1})f(\mathbf{x}); \mathbf{i}; P)\leq \max\{\tau(a_{s}c_{\mathbf{i}-s\mathbf{e}_1}\mathbf{x}^{\mathbf{i}}; \mathbf{i}; P)\}
\]

by Lemma <ref>.
Note that almost all of the coefficients $c_{\mathbf{i}-i\mathbf{e}_1}$ ($i\in \mathbb{Z}$) are $0$.
We also have

\begin{eqnarray*}
\tau(a_{s}c_{\mathbf{i}-s\mathbf{e}_1}\mathbf{x}^{\mathbf{i}}; \mathbf{i}; P)&<&\tau(c_{\mathbf{i}-s\mathbf{e}_1}\mathbf{x}^{\mathbf{i}}; \mathbf{i}; P)-s\val(c_{10})\\
&=&\tau(f; \mathbf{i}-s\mathbf{e}_1; P)+s\mathbf{e}_1\cdot P-s\val(c_{10})\\
&=&\tau(f; \mathbf{i}-s\mathbf{e}_1; P).
\end{eqnarray*}

Hence, we have

\[
\tau(h(\mathbf{x}^{\mathbf{e}_1})f; \mathbf{i}; P)< \tau(f; \overline{\mathbf{i}, \mathbf{i}+\mathbf{e}_1}; P).
\]

For given $f=\sum_{i, j}c_{ij}x^iy^j, g=\sum_{i, j}d_{ij}x^iy^j\in k[x^{\pm1}, y^{\pm1}]$, $L\in \cSI(f, g)$ and an endpoint $P_+\in L$, we consider the following condition.

We have $\Phi_1(L)=\Phi_2(L)=\sembunioil$, $\bio=(0, 0)$, $\bil=(1, 0)$, $\val(c_{00})=\val(d_{00})=0$ and $\val(c_{10})=\val(d_{10})$.
$P_+$ is a vertex of $\Trop(V(f_1))$ ($f_1:=\sum_{i, j}\zeta_{ij}x^iy^j\in \{f, g\}$) and the three vertices of the corresponding $2$-simplex of $\Delta_{f_1}$ are $\bio$, $\bil$ and $\bip=(a, 1)$.
If $L\in \cLSfg$, the other endpoint $P_-\in L$ is a vertex of $\Trop(V(f'_1))$ ($f'_1\in \{f, g\}$) and the three vertices of the corresponding $2$-simplex of $\Delta_{f'_1}$ are $\bio$, $\bil$ and $\bim=(c, -1)$.

Let $f=\sum_{i, j}c_{ij}x^iy^j, g=\sum_{i, j}d_{ij}x^iy^j\in k[x^{\pm1}, y^{\pm1}]$, $L\in \cSI(f, g)$ and an endpoint $P_+\in L$ satisfy the condition $(\P)$.
Let $U$ be a sufficiently small neighborhood of $P_+$, $\overline{L}: x=p_1$ the line containing $L$, and $P\in U\cap (\overline{L}\setminus L)$.
Then, we have

\[
-\val(\zeta_{\bip})=\max\{-\val(c_{\bip}), -\val(d_{\bip})\}.
\]

By Lemma <ref>, for all $\mathbf{i}\in \mathbb{Z}^2\setminus \{\bip\}$, we have

\begin{eqnarray*}
\tau(f_1; \bip; P)>\tau(f; \mathbf{i}; P), \tau(g; \mathbf{i}; P), \tau(f_2; \bip; P)\ (\{f_1, f_2\}=\{f, g\}),
\end{eqnarray*}

and for all $\mathbf{i}\in \mathbb{Z}^2\setminus \{\bio, \bil, \bip\}$, we have

\begin{eqnarray*}
\tau(f_1; \bip; P_+)&=&\tau(f; \bi; P_+)=\tau(g; \bi; P_+)\ (\text{$\bi=\bio$ or $\bil$})\\
&>&\tau(f; \mathbf{i}; P_+), \tau(g; \mathbf{i}; P_+), \tau(f_2; \bip; P_+)\ (f_2\in \{f, g\}\setminus \{f_1\}).
\end{eqnarray*}

Then, from the above inequalities, we have

\[
-\val(\zeta_{\bip})=\max\{-\val(c_{\bip}), -\val(d_{\bip})\}.
\]

Let $f=\sum_{i, j}c_{ij}x^iy^j, g=\sum_{i, j}d_{ij}x^iy^j\in k[x^{\pm1}, y^{\pm1}]$, $L\in \cSI(f, g)$ and an endpoint $P_+\in L$ satisfy the condition $(\P)$.
Assume that the lattice point $\mathbf{i}\in \mathbb{Z}^2$ is in the half plane $\bip+\mathbb{R}\mathbf{e}_1+\mathbb{R}_{\geq 0}\mathbf{e}_2$.
Then, for all $P\in L$, we have

\[
\tau(f_1; \bip; P)> \tau(f_2; \mathbf{i}; P),
\]

\[
\tau(f_1; \bip; P)> \tau(f_1; \mathbf{i}; P)\ \ \text{if $\mathbf{i}\neq \bip$}.
\]

Let $P\in L$ and $f_0=f$ or $g$.
Then, there exists a non-negative number $r\geq0$ such that $P_+=P+r\mathbf{e}_2$, and hence, we have

\begin{eqnarray*}
\tau(f_1; \bip; P)=\tau(f_1; \bip; P_+)-r\bip \cdot \mathbf{e}_2\geq \tau(f_0; \mathbf{i}; P_+)-r.
\end{eqnarray*}

Here, the equality holds only if $f_0=f_1$ and $\mathbf{i}=m_+$.
By the assumption, we have $\mathbf{i}\cdot \mathbf{e}_2\geq1$, and hence, we have

\begin{eqnarray*}
\tau(f_0; \mathbf{i}; P_+)-r=\tau(f_0; \mathbf{i}; P)+r(\mathbf{i}\cdot \mathbf{e}_2)-r\geq \tau(f_0; \mathbf{i}; P).
\end{eqnarray*}

Therefore, we have

\begin{eqnarray*}
\tau(f_1; m_+; P)\geq \tau(f_0; \mathbf{i}; P),
\end{eqnarray*}

and the equality holds only if $f_0=f_1$ and $\mathbf{i}=\bip$.

Let $f=\sum_{i, j}c_{ij}x^iy^j, g=\sum_{i, j}d_{ij}x^iy^j\in k[x^{\pm1}, y^{\pm1}]$, $L\in \cSI(f, g)$ and an endpoint $P_+\in L$ satisfy the condition $(\P)$.
Let $\de \geq 0$ be a non-negative number.
Let $h_{\de}:=h(\de; f; f; \sembunioil)$ and $h'_{\de}:=h(\de; g; f; \sembunioil)$.
Then, for the Laurent polynomials $f_{\de}:=f+h_{\de}(x)f=\sum_{i, j}c'_{ij}x^iy^j$, $g_{\de}:=g+h'_{\de}(x)f=\sum_{i, j}d'_{ij}x^iy^j$ and $G_{\de}:=G(\de; g; f; \sembunioil)=g_{\de}-\frac{d'_{10}}{c'_{10}}f_{\de}=\sum_{i, j}e^{\de}_{ij}x^iy^j$, the following hold.

* $-\val(e^{\de}_{\bip})=-\val(\zeta_{\bip})\ (=\max\{-\val(c_{\bip}), -\val(d_{\bip})\})$.

* Assume that a lattice point $\mathbf{i}\neq \bip$ is in the half plane $\bip+\mathbb{R}\mathbf{e}_1+\mathbb{R}_{\geq 0}\mathbf{e}_2$.
Then, for all $P'\in L$, we have

\[
\tau(G_{\de}; \mathbf{i}; P')<\tau(G_{\de}; \bip; P').
\]

* Assume that $L\in \cLSfg$ and a lattice point $\mathbf{i}\neq \bim$ is in the half plane $\bim +\mathbb{R}\mathbf{e}_1+\mathbb{R}_{\leq 0}\mathbf{e}_2$.
Then, for any point $P'\in L$, we have

\[
\tau(G_{\de}; \mathbf{i}; P')<\tau(G_{\de}; \bim; P').
\]

Let us show (i).
Let $U$ be a sufficiently small neighborhood of $P_+$.
Recall that for a point $P\in U\cap (\Aff(L)\setminus L)$ and for all $\mathbf{i}\in \mathbb{Z}^2\setminus \{\bip\}$, we have

\begin{eqnarray*}
\tau(f_1; \bip; P)>\tau(f; \mathbf{i}; P), \tau(g; \mathbf{i}; P), \tau(f_2; \bip; P),
\end{eqnarray*}

and by Lemma <ref>, we have

\begin{eqnarray*}
\tau(f; \overline{(1, 0)(1, 1)}; P)> \tau(h'_{\de}(x)f; \bip; P),\ \tau(h_{\de}(x)f; \bip; P),
\end{eqnarray*}

Therefore, we have

\[
\tau(f_1; \bip; P)> \tau(h'_{\de}(x)f; \bip; P),\ \tau(h_{\de}(x)f; \bip; P).
\]

Now, we have $\val(\frac{d'_{10}}{c'_{10}})=0$ and

\begin{eqnarray*}
\end{eqnarray*}

Therefore, we have $\tau(G_{\de}; \bip; P)=\tau(f_1; \bip; P)$ by Lemma <ref>.
Thus, we have $-\val(e^{\de}_{\bip})=-\val(\zeta_{\bip})$.

Next, let us show (ii).
(iii) follows from (ii) by symmetry.
Let $P'\in L$.

By Lemmas <ref> and <ref>, we have

\begin{eqnarray*}
\max\{\tau\left(h'_{\de}(x)f; \mathbf{i}; P'\right), \tau\left(-\frac{d'_{10}}{c'_{10}}h_{\de}f; \mathbf{i}; P'\right)\}< \tau(f; \overline{\bi, \mathbf{i}+\mathbf{e}_1}; P') \leq \tau(f_1; \bip; P').
\end{eqnarray*}

Since $\mathbf{i}\neq \bip$, by Lemma <ref>, we have

\begin{eqnarray*}
\max\{\tau(g; \mathbf{i}; P'), \tau\left(-\frac{d'_{10}}{c'_{10}}f; \mathbf{i}; P'\right)\}<\tau(f_1; \bip; P').
\end{eqnarray*}

Therefore, by Lemma <ref>, we have

\begin{eqnarray*}
&&\tau(G_{\de}; \mathbf{i}; P')\\
&\leq& \max\{\tau(g; \mathbf{i}; P'), \tau(h'_{\de}(x)f; \mathbf{i}; P'), \tau\left(-\frac{d'_{10}}{c'_{10}}f; \mathbf{i}; P'\right),\tau\left(-\frac{d'_{10}}{c'_{10}}h_{\de}f; \mathbf{i}; P'\right)\}\\
&<&\tau(f_1; \bip; P').
\end{eqnarray*}

Since $\val(\zeta_{\bip})=\val(e^{\de}_{\bip})$, we have

\[
\tau(f_1; \bip; P')=\tau(G_{\de}; \bip; P').
\]

Thus, we have

\[
\tau(G_{\de}; \mathbf{i}; P')<\tau(G_{\de}; \bip; P').
\]

§ PROOFS OF THE MAIN THEOREMS

The following proposition and its corollary show a special case of the main theorems where $\cSI'$ consists of one element.
The arguments of the proof will play an important part in the proofs of the main theorems.

Let $f=\sum_{i, j}c_{ij}x^iy^j, g=\sum_{i, j}d_{ij}x^iy^j\in k[x^{\pm1}, y^{\pm1}]$, $L\in \cSI(f, g)$ and an endpoint $P_+:=(p_1, y_+)\in L$ satisfy the condition $(\P)$.
For a non-negative number $\de \geq 0$, we write $G_{\de}:=G(\de; g; f; \sembunioil)=\sum_{i, j}e^{\de}_{ij}x^iy^j$.

* Assume $L\in \cRayfg$.
If for any $\de \geq 0$, we have $\val(e^{\de}_{00})>\de$, then $\trop(V(f, g))\cap L=\emptyset$.
Otherwise, there exists $\de_0 \geq 0$ with $\val(e^{\de_0}_{00})\leq \de_0$.
In this case, $\de_m:=\val(e^{\de_0}_{00})=\min\{\de \in \mathbb{R}_{\geq 0}\ |\ \val(e^{\de}_{00})\leq \de\}$, and for any $\de \geq \de_m$, we have

\[
\trop(V(f, g))\cap L=\Trop(V(G_{\de}))\cap L=\{(p_1, y_+-\de_m))\}.
\]

* Assume $L\in \cLSfg$.
Let $\de'>0$ be the half of the lattice length of $L$.
Then we have

\begin{eqnarray*}
&&\trop(V(f, g))\cap L=\Trop(V(G_{\de}))\cap L\\
\begin{cases}
\{(p_1, y_+-\val(e^{\de'}_{00})), (p_1, y_-+\val(e^{\de'}_{00}))\} & (\val(e^{\de'}_{00})< \de'),\\
\{(p_1, y_+-\de')\}\ (\text{multiplicity $=2$}) & (\val(e^{\de'}_{00})\geq \de').
\end{cases}
\end{eqnarray*}

For a non-negative number $\de\geq 0$, we write $f_{\de}:=f+h_{\de}(x)f=\sum_{i, j}c'_{ij}x^iy^j$ and $g_{\de}:=g+h'_{\de}(x)f=\sum_{i, j}d'_{ij}x^iy^j$.

Note that for $\de\geq 0$ and $G_{\de}=\sum_{i, j}e^{\de}_{ij}x^iy^j$, we have $e^{\de}_{10}=0$ and $\val(e^{\de}_{00})=\val\left(d'_{00}-\frac{d'_{10}}{c'_{10}}c'_{00}\right)\geq 0$.

Let us show (i).
Let $L\in \cRay(f, g)$.

By the assumption that the three vertices of the corresponding $2$-simplex of $\Delta_{f_1}$ are $\bio$, $\bil$ and $\bip=(a, 1)$, we have $L=P_++\mathbb{R}_{\leq 0}\mathbf{e}_2$, and this implies

\[
\forall (i, j)\in \mathbb{Z}^2, j<0\Rightarrow c_{ij}=d_{ij}=0.
\]

Combined with Lemma <ref>, we have the following claim.

For any $\de\geq 0$, we have

\[
\{\mathbf{i}\in \mathbb{Z}^2\ |\ \exists P\in L\text{ s.t. }\tau(G_{\de}; \mathbf{i}; P)=\trop(G_{\de})(P)\} \subset \mathbb{Z}\mathbf{e}_1\cup \{\bip\}.
\]

For any $\de\geq 0$ and $P\in L$, we have

\[
\forall n\in \mathbb{Z}\setminus \{0\}, \tau(G_{\de}; (n, 0); P)<-\de.
\]

Let $\de\geq 0$ and $P\in L$.
Since we have

\begin{eqnarray*}
\forall n\in \mathbb{Z}\setminus \{0, 1\}, &\val(c'_{n0})>n\val(c_{10})+\de,& \\
\end{eqnarray*}

and $G_{\de}=\sum_{i, j}e^{\de}_{ij}x^iy^j=g_{\de}-\frac{d'_{10}}{c'_{10}}f_{\de}$, we have

\begin{eqnarray*}
\forall n\in \mathbb{Z}\setminus \{0, 1\}, \val(e^{\de}_{n0})>n\val(c_{10})+\de.
\end{eqnarray*}

From $\tau(f; (0, 0); P)=\tau(f; (1, 0); P)$, it follows that $\val(c_{10})=\mathbf{e}_1\cdot P=p_1$.
Therefore, we have

\[
\forall n\in \mathbb{Z}\setminus \{0, 1\}, \val(e^{\de}_{n0})>n\mathbf{e}_1\cdot P+\de,
\]

\[
\forall n\in \mathbb{Z}\setminus \{0, 1\}, \tau(G_{\de}; (n, 0); P)<-\de.
\]

Combined with

\begin{eqnarray*}
\tau(G_{\de}; (1, 0); P)=-\val(e^{\de}_{10})+\mathbf{e}_1\cdot P=-\infty<-\de,
\end{eqnarray*}

we see that

\[
\forall n\in \mathbb{Z}\setminus \{0\}, \tau(G_{\de}; (n, 0); P)<-\de.
\]

For any $\de\geq 0$, we have

\begin{eqnarray*}
\Trop(V(G_{\de}))\cap \{(p_1, y)\ |\ y_+-\de \leq y\leq y_+\}\\
\begin{cases}
\{(p_1, y_+-\val(e^{\de}_{00}))\} & (\val(e^{\de}_{00})\leq \de),\\
\emptyset & (\val(e^{\de}_{00})>\de).
\end{cases}
\end{eqnarray*}

Let $\de\geq 0$.
By Lemma <ref>, for a point $(p_1, t)\in L$, we have

\begin{eqnarray*}
\tau(G_{\de}; \bip; (p_1, t))&=&\tau(f_1; \bip; (p_1, y_++(t-y_+)))\\
&=&\tau(f_1; \bip; P_+)+(t-y_+)\bip \cdot \mathbf{e}_2\\
&=&\tau(f_1; (0, 0); P_+)+(t-y_+)\\
\end{eqnarray*}

Hence, we have

\begin{eqnarray*}
y_+-\val(e^{\de}_{00})<t\leq y_+&\Rightarrow& \tau(G_{\de}; \bip; (p_1, t))> \tau(G_{\de}; (0, 0); (p_1, t)),\\
t=y_+-\val(e^{\de}_{00})&\Rightarrow& \tau(G_{\de}; \bip; (p_1, t))=\tau(G_{\de}; (0, 0); (p_1, t)),\\
t<y_+-\val(e^{\de}_{00})&\Rightarrow& \tau(G_{\de}; \bip; (p_1, t))<\tau(G_{\de}; (0, 0); (p_1, t)).
\end{eqnarray*}

Thus, if $\val(e^{\de}_{00})>\de$, then we have

\begin{eqnarray*}
\Trop(V(G_{\de}))\cap \{(p_1, y)\ |\ y_+-\de \leq y\leq y_+\}=\emptyset.
\end{eqnarray*}

If $\val(e^{\de}_{00})\leq \de$, then, by Claim <ref>, for any point $P\in L$ and $n\in \mathbb{Z}\setminus \{0\}$, we have

\[
\tau(G_{\de}; (n, 0); P)<-\de \leq -\val(e^{\de}_{00})=\tau(G_{\de}; (0, 0); P),
\]

and hence, by Claim <ref>, we have

\begin{eqnarray*}
\Trop(V(G_{\de}))\cap \{(p_1, y)\ |\ y_+-\de \leq y\leq y_+\}=\{(p_1, y_+-\val(e^{\de}_{00}))\}.
\end{eqnarray*}

Let us conclude the proof of (i).
Assume that for any $\de \geq 0$, we have $\val(e^{\de}_{00})>\de$.
Let $\de>0$ be a positive number.
Then, we have $\val(e^{\de}_{00})>\de$, and hence, for any point $P\in L$ and $n\in \mathbb{Z}$, we have

\[
\tau(G_{\de}; (n, 0); P)<-\de.
\]

Then, for all $n\in \mathbb{Z}$, we have

\begin{eqnarray*}
y_+-\de<t\leq y_+\Rightarrow \tau(G_{\de}; \bip; (p_1, t))=t-y_+>-\de \geq \tau(G_{\de}; (n, 0); (p_1, t)).
\end{eqnarray*}

Thus, by Claim <ref>, we have

\[
\Trop(V(G_{\de}))\cap \{(p_1, t)\ |\ y_+-\de <t\leq y_+\}=\emptyset.
\]

Combined with

\begin{eqnarray*}
G_{\de}=g_{\de}-\frac{d'_{10}}{c'_{10}}f_{\de}=g+\left(h'_{\de}(x)-\frac{d'_{10}}{c'_{10}}(1+h_{\de}(x))\right)f\in (f, g),
\end{eqnarray*}

it follows that $\trop(V(f, g))\subset \Trop(V(G_{\de}))$.
Hence, for any $\de \geq 0$, we have

\[
\trop(V(f, g))\cap \{(p_1, t)\ |\ y_+-\de <t\leq y_+\}=\emptyset.
\]

Thus, we have $\trop(V(f, g))\cap L=\emptyset$.

Next, assume that there exists a non-negative number $\de_0 \geq 0$ such that $\val(e^{\de_0}_{00})\leq \de_0$.
First, consider the case where $\val\left(d_{00}-\frac{d_{10}}{c_{10}}c_{00}\right)=0$.
For $\de=0$, we have $h_{0}(t)=h'_{0}(t)=0$ and $G_{0}=g-\frac{d_{10}}{c_{10}}f$, and hence, we have $e^{0}_{00}=d_{00}-\frac{d_{10}}{c_{10}}c_{00}$.
Thus, we have

\[
\val(e^{0}_{00})=\val\left(d_{00}-\frac{d_{10}}{c_{10}}c_{00}\right)=0,
\]

and hence, by Claim <ref>, we have

\[
\Trop(V(G_0))\cap L=\{(p_1, y_+-0)\}=\{P_+\}.
\]

Let $U$ be a sufficiently small neighborhood of $P_+$.
Then, we have

\begin{eqnarray*}
\Trop(V(G_0))\cap \Trop(V(f_2))\cap U&=&\Trop(V(G_0))\cap L\cap U\ \ (\text{see Figure \ref{Fig_G_delta}})\\
\end{eqnarray*}

and hence, the point $\{P_+\}$ is a connected component of $\Trop(V(f_2)) \cap \Trop(V(G_0))$.
Note that the intersection multiplicity at $P_+\in \Trop(V(f_2)) \cap \Trop(V(G_0))$ is $1$ (see Figure <ref>).
Hence, by Theorem <ref>, there exists a unique point $\mathbf{x}_+\in V(f_2, G_0)$ such that $\trop(\mathbf{x}_+)=P_+$.
Note that since we have $G_0=g-\frac{d_{10}}{c_{10}}f$, we have

\[
(f, G_0)=(g, G_0)=(f, g).
\]

Thus, we have

\[
P_+\in \trop(V(f_2, G_0))=\trop(V(f, g)).
\]

Since we have $G_0\in (f, g)$, we have $V(f, g)\subset V(G_0)$, and hence, we have

\[
\trop(V(f, g))|_L\subset \{P_+\}.
\]

Thus, we have

\[
\trop(V(f, g))|_L=\{P_+\}.
\]

Next, we consider the case where $\de_0 > 0$.
Let $\de_m=\val(e^{\de_0}_{00})$.
By Claim <ref>, we have

\[
\Trop(V(G_{\de_0}))\cap L=\{(p_1, y_+-\de_m)\}.
\]

Let $U_L$ be a sufficiently small neighborhood of the ray $L$.
Then, we have

\begin{eqnarray*}
\Trop(V(G_{\de_0}))\cap \Trop(V(f))\cap U_L&=&\Trop(V(G_{\de_0}))\cap L\ \ (\text{see Figure \ref{Fig_G_delta}})\\
&=&\{(p_1, y_+-\de_m)\}.
\end{eqnarray*}

Hence, $\{(p_1, y_+-\de_m)\}$ is a connected component of $\Trop(V(f)) \cap \Trop(V(G_{\de_0}))$.
Note that the intersection multiplicity at $(p_1, y_+-\de_m)\in \Trop(V(f)) \cap \Trop(V(G_{\de_0}))$ is $1$ (see Figure <ref>).
Hence, by Theorem <ref>, there exists a unique point $\mathbf{x}\in V(f, G_{\de_0})$ such that $\trop(\mathbf{x})=(p_1, y_+-\de_m)$.

Thus, we have

\[
(p_1, y_+-\de_m)\in \trop(V(f, G_{\de_0}))=\trop(V(f, g)).
\]

From $V(f, g)\subset V(G_{\de_0})$, it follows that

\[
\trop(V(f, g))\cap L\subset \{(p_1, y_+-\de_m)\}.
\]

Thus, we have

\[
\trop(V(f, g))\cap L=\{(p_1, y_+-\de_m)\}.
\]

In the following, we permit the case where $\de_m=0$, i.e., $\val(e^{\de_0}_{00})=0$.
For any $\de \geq 0$, we have $G_{\de}\in (f, g)$, and hence,

\[
\{(p_1, y_+-\de_m)\}=\trop(V(f, g))\cap L\subset \Trop(V(G_{\de}))\cap L.
\]

If $\de<\de_m$ and $\val(e^{\de}_{00})\leq \de$, then by Claim <ref>, $\trop(V(f, g))\cap L$ must contain $(p_1, y_+-\val(e^{\de}_{00}))$, a contradiction.

For any $\de \geq \de_m$, $\Trop(V(G_{\de}))\cap L$ must contain $(p_1, y_+-\de_m)$, combined with Claim <ref>, we have $\val(e^{\de}_{00})=\de_m$.
Thus, $\de_m=\min\{\de \in \mathbb{R}_{\geq 0}\ |\ \val(e^{\de}_{00})\leq \de\}$, and for any $\de \geq \de_m$, we have

\[
\trop(V(f, g))\cap L=\Trop(V(G_{\de}))\cap L=\{(p_1, y_+-\de_m))\}.
\]

(-0.7,-0.75) circle [radius=0.275];
(0.7,-0.75) circle [radius=0.275];
(0,0.6) circle [radius=0.275];

(3.3,-0.75) circle [radius=0.275];
(4.7,-0.75) circle [radius=0.275];

(7.75,-1.85) circle [radius=0.275];
(7.75,-0.7) circle [radius=0.275];
(8,0) to [out=0, in=90] (8.5,-0.45);
(8,-1.4) to [out=0, in=-90] (8.5,-0.95);
[label=right:$\val(e^{\de}_{00})$] (x1) at (8.2,-0.7);

(5.2,0) arc (0:180:1.2);
(A11) at (4,0);
[label=right:$P_+$] (w1) at (3.95,0);
(A12) at (4,0.3);
(A13) at (4,-3);
[very thick] (A13)–(4,1.2);
[label=above:$\bio$] (a2) at (3.3,-0.99);
[label=above:$\bil$] (a3) at (4.7,-0.99);

(1.2,0) arc (0:180:1.2);
(9.2,0) arc (0:180:1.2);

[very thick] (6.8,-0.8)–(9.2,-2);

(A1) at (0,0);
[label=right:$P_+$] (B1) at (-0.05,-0.2);
(A2) at (0,0.3);
(A3) at (0,-3);
(A4) at (8,0);
[label=above:$P_+$] (C1) at (8,0);
(A5) at (8,0.3);
(A6) at (8,-3);
[label=right:$L$] (A7) at (0,-2.6);
[label=right:$L$] (A8) at (8,-2.6);

[label=below:$U_L$] (B4) at (-1,1.4);
[label=below:$U_L$] (B5) at (3,1.4);
[label=below:$U_L$] (B5) at (7,1.4);

[label=above:$\bip$] (a1) at (0.01,0.32);
[label=above:$\bio$] (a2) at (-0.68,-0.99);
[label=above:$\bil$] (a3) at (0.7,-0.99);
[label=above:$\bio$] (a4) at (7.77,-2.09);
[label=above:$\bip$] (a5) at (7.76,-0.97);

[very thick] (A1)–(A3);
[very thick] (-1.039,0.6)–(A1)–(1.039,0.6);
[dotted, thick] (A4)–(8,-0.58);
[dotted, thick] (8,-0.86)–(8,-1.675);
[dotted, thick] (8,-2.05)–(8,-3);

in 1,4,11 [black] (A) circle (0.06);
[label=below:$\Trop(V(f_1))$] (a) at (0,-3.3);
[label=below:$\Trop(V(f_2))$] (a) at (4,-3.3);
[label=below:$\Trop(V(G_{\de}))$] (b) at (8,-3.3);

$\Trop(V(f_1))$, $\Trop(V(f_2))$ and $\Trop(V(G_{\de}))$ in a neighborhood $U_L$ of $L$.

Next, let us show (ii).
Let $L\in \cLSfg$.
Let $P_-=(p_1, y_-)$ ($y_-<y_+$) be the other endpoint of the line segment $L$ and $\de':=\dist(P_+, P_-)/2=(y_+-y_-)/2$.
Let $\sigma_-$ be the corresponding $2$-simplex of $\Delta_{f'_1}$ ($f'_1\in \{f, g\}$), and $\bim=(c, -1)$ the vertex of $\sigma_-$ other than $\bio$ and $\bil$.

Note that $L=\{(p_1, t)\ |\ y_-\leq t\leq y_+\}$.
As in the case of $L\in \cRay(f, g)$, we have

\begin{eqnarray*}
\tau(G_{\de'}; \bip; (p_1, t))=t-y_+,\ \ \tau(G_{\de'}; \bim; (p_1, t))=y_--t.
\end{eqnarray*}

We have

\[
\{\mathbf{i}\in \mathbb{Z}^2\ |\ \exists P\in L\text{ s.t. }\tau(G_{\de'}; \mathbf{i}; P)=\trop(G_{\de'})(P)\} \subset \{\bio, \bip, \bim\}.
\]

Let $\mathbf{i}'$ be an element of the left hand side different from $\bip$, $\bim$.
Then, there exists a point $P=(p_1, t)\in L$ satisfying $\tau(G_{\de'}; \mathbf{i}'; P)=\trop(G_{\de'})(P)$.
Here, by Lemma <ref>, we have $\mathbf{i}' \in \mathbb{Z}\mathbf{e}_1$.
As in the case of $L\in \cRay(f, g)$, we have

\[
\forall n\in \mathbb{Z}\setminus \{0\}, \tau(G_{\de'}; (n, 0); P)<-\de'.
\]

Since we have $y_-\leq t\leq y_+$, we have

\[
\max\{\tau(G_{\de'}; \bip; P), \tau(G_{\de'}; \bim; P)\}=\max\{t-y_+, y_--t\}\geq \frac{y_--y_+}{2}=-\de',
\]

and hence, we have

\[
\forall n\in \mathbb{Z}\setminus \{0\}, \tau(G_{\de'}; (n, 0); P)<\max\{\tau(G_{\de'}; \bip; P), \tau(G_{\de'}; \bim; P)\}.
\]

Thus, we have

\[
\mathbf{i}' =(0, 0).
\]

Let us conclude the proof of (ii).

Consider the case where $0\leq \val(e^{\de'}_{00})<\de'$.
Let $y_1:=y_+-\val(e^{\de'}_{00})$ and $y_2:=y_-+\val(e^{\de'}_{00})$.
Then, we have

\begin{eqnarray*}
y_1<t\leq y_+&\Rightarrow& \tau(G_{\de'}; \bip; (p_1, t))> \tau(G_{\de'}; \bim; (p_1, t)),\ \tau(G_{\de'}; \bio; (p_1, t)),\\
t=y_1&\Rightarrow& \tau(G_{\de'}; \bip; (p_1, t))=\tau(G_{\de'}; \bio; (p_1, t))>\tau(G_{\de'}; \bim; (p_1, t)),\\
y_2< t<y_1&\Rightarrow& \tau(G_{\de'}; \bio; (p_1, t))> \tau(G_{\de'}; \bip; (p_1, t)),\ \tau(G_{\de'}; \bim; (p_1, t)),\\
t=y_2&\Rightarrow& \tau(G_{\de'}; \bim; (p_1, t))=\tau(G_{\de'}; \bio; (p_1, t))> \tau(G_{\de}; \bip; (p_1, t)),\\
y_-\leq t<y_2&\Rightarrow& \tau(G_{\de'}; \bim; (p_1, t))> \tau(G_{\de}; \bio; (p_1, t)),\ \tau(G_{\de'}; \bip; (p_1, t)).
\end{eqnarray*}

Therefore, by Claim <ref>, we have

\[
\Trop(V(G_{\de'}))\cap L=\{(p_1, y_1), (p_1, y_2)\}=\{(p_1, y_+-\val(e^{\de'}_{00})), (p_1, y_-+\val(e^{\de'}_{00}))\}.
\]

Then, as in the case of $L\in \cRay(f, g)$, we have

\[
\trop(V(f, g))\cap L=\{(p_1, y_1), (p_1, y_2)\}=\{(p_1, y_+-\val(e^{\de'}_{00})), (p_1, y_-+\val(e^{\de'}_{00}))\}.
\]

Next, consider the case where $\val(e^{\de'}_{00})\geq \de'$.
Then, we have

\begin{eqnarray*}
\frac{y_++y_-}{2}<t\leq y_+&\Rightarrow& \tau(G_{\de'}; \bip; (p_1, t))> \tau(G_{\de'}; \bim; (p_1, t)),\ \tau(G_{\de'}; \bio; (p_1, t)),\\
t=\frac{y_++y_-}{2}&\Rightarrow& \tau(G_{\de'}; \bip; (p_1, t))=\tau(G_{\de'}; \bim; (p_1, t))\geq \tau(G_{\de'}; \bio; (p_1, t)),\\
y_-\leq t<\frac{y_++y_-}{2}&\Rightarrow& \tau(G_{\de'}; \bim; (p_1, t))> \tau(G_{\de'}; \bio; (p_1, t)),\ \tau(G_{\de'}; \bip; (p_1, t)).
\end{eqnarray*}

Therefore, by Claim <ref>, we have

\[
\Trop(V(G_{\de'}))\cap L=\{\left(p_1, \frac{y_++y_-}{2}\right)\}.
\]

Recall that $m_+=(a, 1)$ and $m_-=(c, -1)$.
Then, by Remark <ref>, we have

\[
i(\left(p_1, \frac{y_++y_-}{2}\right); \Trop(V(f)) \cdot \Trop(V(G_{\de'})))
1 & a-c \\
0 & 2 \\
\end{pmatrix}\right|
\]

Then, as in the case of $L\in \cRay(f, g)$, we have

\[
\trop(V(f, g))\cap L=\{\left(p_1, \frac{y_++y_-}{2}\right)\}\ \ (\text{multiplicity}=2).
\]

Thus, we conclude the proof of Proposition <ref>.

The following corollary is clear.

Let $f, g\in k[x^{\pm1}, y^{\pm1}]$ be Laurent polynomials and $L\in \cRay(f, g)$ a ray.
Then, there are at most one point in the intersection $\trop(V(f, g))\cap L$.

The following corollary shows a special case of the main theorems where $\cSI'$ consists of one element.
The arguments of the proof will play an important part in the proofs of the main theorems.

Let $f=\sum_{i, j}c_{ij}x^iy^j$ and $g=\sum_{i, j}d_{ij}x^iy^j$ be Laurent polynomials in $k[x^{\pm1}, y^{\pm1}]$ and $D$ a divisor on $\Trop(V(f))$ such that there is a tropical rational function $\psi$ satisfying $(*)$ and $(\psi)=D-E$.
Let $L\in \cSI(f, g)$ be a ray or a line segment and let $\Phi_2(L)=\sembunioil$.
Then there exists an element $\tilde{d}_{\bio}\in k$ such that if we set $g':=g-d_{\bio}{\mathbf{x}}^{\bio}+\tilde{d}_{\bio}{\mathbf{x}}^{\bio}$, we have $\trop(g)=\trop(g')$ and $\trop(V(f, g'))|_L=D|_L$.

We will show the case where $L\in \cRay(f, g)$, and the case where $L\in \cLSfg$ can be shown similarly.
By applying an affine transformation and multiplying units in $k[x^{\pm1}, y^{\pm1}]$ to $f$ and $g$, we may assume that $f$, $g$, $L$ and an endpoint $P_+:=(p_1, y_+)\in L$ satisfy the condition $(\P)$.
Note that multiplying units in $k[x^{\pm1}, y^{\pm1}]$ to $f$ and $g$ does not change $V(f, g)$ and hence $\Trop(V(f, g))$.

Let $P_1=(p_1, y_+-\de)$ ($\de \geq 0$) be the intersection point of $D\cap L$.
Recall that $P_+$ is a vertex of $\Trop(V(f_1))$ and $f_1=\sum_{i, j}\zeta_{ij}x^iy^j$.

Since we have $\tau(f; \bio; P_+)=\tau(f_1; \bip; P_+)$ and $P_1=P_+-\de \mathbf{e}_2$,
we have

\begin{eqnarray*}
\tau(f_1; \bip; P_1)=\tau(f_1; \bip; P_+)-\de (\bip \cdot \mathbf{e}_2)=\tau(f_1; (0, 0); P_+)-\de=-\de.
\end{eqnarray*}

Thus, we have

\[
\de=-\tau(f_1; \bip; P_1)=\val(\zeta_{\bip})-\bip \cdot P_1.
\]

Since the coordinates of $P_1$ are assumed to belong to the value group, there exists an $\alpha\in k^*$ such that $\val(\alpha)=\val(\zeta_{\bip})-\bip \cdot P_1=\de$.

First, consider the case of $\de=0$.
By Theorem <ref>, there exists an element $\mathbf{t}=(t_1, t_2)\in (k^*)^2$ such that $\trop(\mathbf{t})=\mathbf{0}$, and $\trop(V(f)\cap \mathbf{t}V(g))=E$.

Then, for the polynomial $g\left(\frac{\mathbf{x}}{\mathbf{t}}\right)$, we have

\[
\trop(V(f, g\left(\frac{\mathbf{x}}{\mathbf{t}}\right)))=E.
\]

Note that we have $\trop(g\left(\frac{\mathbf{x}}{\mathbf{t}}\right))=\trop(g)$.
By Proposition <ref>, we have

\[
\val\left(d_{00}-\frac{d_{10}}{t_1c_{10}}c_{00}\right)=0,
\]

and hence, we have

\[
\val\left(t_1d_{00}-\frac{d_{10}}{c_{10}}c_{00}\right)=0.
\]

We set $g_0:=g-d_{00}+t_1d_{00}$.
Then, $\trop(g_0)=\trop(g)$, and by Proposition <ref>, we have $\trop(V(f, g_0))|_L=E|_L$.

Next, consider the case of $\de>0$.
Let $h_{\de}:=h(\de; f; f; \sembunioil), h'_{\de}:=h(\de; g; f; \sembunioil)$, $f_{\de}:=f+h_{\de}(x)f=\sum_{i, j}c'_{ij}x^iy^j$ , $g_{\de}:=g+h_{\de}(x)f=\sum_{i, j}d'_{ij}x^iy^j$ and $G_{\de}:=g_{\de}-\frac{d'_{10}}{c'_{10}}f_{\de}=\sum_{i, j}e_{ij}x^iy^j$.
By the construction of $g_{\de}$, there exists a term $\beta\in k$ such that $d'_{00}=d_{00}+\beta$ and $\val(\beta)>0$.
We set

\begin{equation*}
\tilde{d}_{00}=\alpha-\beta+\frac{d'_{10}}{c'_{10}}c'_{00}.
\end{equation*}

Since we have

\begin{eqnarray*}
\val(\alpha)=\de>0,\ \val(\beta)>0,\ \val\left(\frac{d'_{10}}{c'_{10}}c'_{00}\right)=0,
\end{eqnarray*}

we have $\val(\tilde{d}_{00})=0=\val(d_{00})$.
We set $g':=g-d_{00}+\tilde{d}_{00}$.
Then, we have $\trop(g')=\trop(g)$.
Note that

\[
h(\de; g'; f; \sembunioil)=h(\de; g-d_{00}+\tilde{d}_{00}; f; \sembunioil)=h(\de; g; f; \sembunioil)=h'_{\de}.
\]

For the Laurent polynomial

\[
G'_{\de}:=g'+h'_{\de}(x)f-\frac{d'_{10}}{c'_{10}}f_{\de}=G_{\de}-d_{00}+\tilde{d}_{00}=\sum_{i, j}e'_{ij}x^iy^j,
\]

we have $e'_{00}=\alpha$ and $e'_{\bi}=e_{\bi}$ ($\bi \neq (0, 0)$).
Here, we have $\val(e'_{00})=\val(\alpha)=\de$, and hence, by Proposition <ref>, we have $\trop(V(f, g'))|_L=D|_L$.

In Corollary <ref>, we change the coefficient $d_{\bio}$.
By symmetry, however, we can change the coefficient $d_{\bil}$ instead of changing the coefficient $d_{\bio}$.

Let $f=\sum_{i, j}c_{ij}x^iy^j$, $g=\sum_{i, j}d_{ij}x^iy^j$ and $g_2=\sum_{i, j}d_{2, ij}x^iy^j$ be Laurent polynomials in $k[x^{\pm1}, y^{\pm1}]$ such that $\trop(g)=\trop(g_2)$, and $D$ a divisor on $\Trop(V(f))$ satisfying $(*)$ and $(\psi)=D-E$.
Let $L\in \cSI(f, g)=\cSI(f, g_2)$ be a ray or a line segment, $\Phi_2(L)=\sembunioil$ and $\de=\dist(D|_L, E|_L)$.
Let $l=\Aff(\sembunioil)$ and

\[
l_{\de}:=\{\mathbf{i}_0+n(\mathbf{i}_1-\mathbf{i}_0) \in \mathbb{Z}^2\cap l\ |\ \mu_n(g; \sembunioil)\leq \de\}.
\]

Assume that $\Trop(V(f, g))|_L=D|_L$ and for all $\mathbf{i}\in l_{\de}$, $d_{\mathbf{i}}=d_{2, \mathbf{i}}$.
Then, we have

\[
\trop(V(f, g_2))|_L=\trop(V(f, g))|_L.
\]

By applying an affine transformation and multiplying units in $k[x^{\pm1}, y^{\pm1}]$ to $f$ and $g$, we may assume that $f$, $g$, $L$ and an endpoint $P_+:=(p_1, y_+)\in L$ satisfy the condition $(\P)$.
We have $\trop(g)=\trop(g_2)$, and hence, $f$, $g_2$, $L$ and $P_+$ also satisfy the condition $(\P)$.

Since $d_{\mathbf{t}}=d_{2, \mathbf{t}}$ ($\forall \mathbf{t}\in l_{\de})$, we have $h(\de; g; f; \sembunioil)=h(\de; g_2; f; \sembunioil)$.
Let $h_{\de}:=h(\de; f; f; \sembunioil)$, $h'_{\de}:=h(\de; g; f; \sembunioil)=h(\de; g_2; f; \sembunioil)$.

Let $f_{\de}:=f+h_{\de}(x)f=\sum_{i, j}c'_{ij}x^iy^j$, $g_{\de}:=g+h'_{\de}(x)f=\sum_{i, j}d'_{ij}x^iy^j$ and $g_{2, \de}:=g_2+h'_{\de}(x)f=\sum_{i, j}d''_{ij}x^iy^j$.
Then, we have $d'_{00}=d''_{00}$ and $d'_{10}=d''_{10}$.
For the Laurent polynomials

\begin{eqnarray*}
G_{\de}&:=&g_{\de}-\frac{d'_{10}}{c'_{10}}f_{\de}=\sum_{i, j}e_{ij}x^iy^j,\\
G_{2, \de}&:=&g_{2, \de}-\frac{d'_{10}}{c'_{10}}f_{\de}=\sum_{i, j}e'_{ij}x^iy^j,
\end{eqnarray*}

we have $e_{00}=e'_{00}$, and hence, $val(e_{00})=\val(e'_{00})=\de$.

Thus, by Proposition <ref>, we have

\[
\trop(V(f, g_2))|_L=\trop(V(f, g))|_L.
\]

Let $f$ and $g=\sum_{i, j}d_{ij}x^iy^j$ be Laurent polynomials in $k[x^{\pm1}, y^{\pm1}]$ and $D$ a divisor on $\Trop(V(f))$ satisfying $(*)$ and $(\psi)=D-E$.
Let $\cSI'$ be a subset of the union of $\cSI(f, g)$ and $\cPI:=\cPI(f, g)$.
Assume that $\cSI'$ is acyclic with respect to $\Phi_2$ and that for each $L\in \cSI'$, we have $\dist(D|_L, E|_L)<\mu(g; \sembunij)$ $(\sembunij:=\Phi_g(L))$.
Then, there exists $g'\in k[x^{\pm1}, y^{\pm1}]$ such that $\trop(g')=\trop(g)$ and

\[
\trop(V(f, g'))|_{\cSI' \cup \cPI}=D|_{\cSI' \cup \cPI}.
\]

Let $\mathcal{C}$ be the union of the elements of $\Delta':=\Phi_2(\cSI')$.
We number and order the endpoints of the elements of $\mathcal{C}$ as $p_1< \dots < p_n$ so that this ordering is normal on each tree of the forest.
We set $\tilde{d}_{\mathbf{i}}=d_{\mathbf{i}}$ for all $\mathbf{i}\in \mathbb{Z}^2\setminus \{p_1, \dots, p_n\}$.
We will construct $g'=\sum_{\mathbf{i}}\tilde{d}_{\mathbf{i}}\mathbf{x}^{\mathbf{i}}$ by determining the coefficients $\tilde{d}_{p_{i}}$ ($i=1, \dots, n$) inductively.
First, we set $\tilde{d}_{p_1}=d_{p_1}$.
Assume that we determine the coefficients $\tilde{d}_{p_i}$ ($1, \dots, t-1$).
Let $g_{t-1}=g-\sum_{i=1}^{t-1}d_{p_i}\mathbf{x}^{p_i}+\sum_{i=1}^{t-1}\tilde{d}_{p_i}\mathbf{x}^{p_i}$, $T$ be the component of $\mathcal{C}$ containing $p_t$, and $m=\min \{i\in \mathbb{Z}\ |\ p_i\in T\}$.
Note that the restriction of the ordering $p_1< \dots < p_n$ to $T$ is normal by the assumption.
If $t=m$, we set $\tilde{d}_{p_t}=d_{p_t}$.
If $t>m$, the unique path $p_mTp_t$ contains the element $\sembunpspt$ of $\Delta'$.
Let $L_{st}\in \cSI'$ be the ray or the line segment corresponding to $\sembunpspt$.
Note that the $\Aff(\sembunpspt)$ might contain another point $p_r$ ($r>\max\{s, t\}$).
By the assumption, however, we have $\dist(D|_{L_{st}}, E|_{L_{st}})<\mu(g_{t-1}; \sembunpspt)$.
By Corollary <ref>, we determine an element $\tilde{d}_{p_t}\in k$.
In this case, we set $g_t=g_{t-1}-d_{p_t}\mathbf{x}^{p_t}+\tilde{d}_{p_t}\mathbf{x}^{p_t}$, then we have $\val(\tilde{d}_{p_t})=\val(d_{p_t})$ and $\trop(V(f, g_t))|_{L_{st}}=D|_{L_{st}}$.
Note that we can change the coefficient $d_{p_t}$ by Remark <ref>.

By repeating this process, we get the Laurent polynomial $g'=g-\sum_{i=1}^{n}d_{p_i}\mathbf{x}^{p_i}+\sum_{i=1}^{n}\tilde{d}_{p_i}\mathbf{x}^{p_i}$.
Corollary <ref> shows that for all $L\in \cSI'$, we have

\[
\trop(V(f, g'))|_L=D|_L.
\]

Since we have $\trop(g')=\trop(g)$, we have $\cPI(f, g)=\cPI(f, g')\subset \trop(V(f, g'))$ with the multiplicities taken into account by Theorem <ref>.
Thus, we conclude the proof of Theorem <ref>.

Let $f, g\in k[x^{\pm1}, y^{\pm1}]$ be Laurent polynomials and $D$ a divisor on $\Trop(V(f))$ satisfying $(*)$ and $(\psi)=D-E$.
Let $\cSI'$ be a subset of the union of $\cSI(f, g)$ and $\cPI:=\cPI(f, g)$.
Assume that $\cSI'$ is acyclic with respect to $\Phi_2$ and that we can number and order the endpoints of the elements of $\Delta':=\Phi_2(\cSI')$ as $p_1< \dots < p_n$ so that this order is normal on each tree of the forest.
Assume that for each element $\sembunpipj$ of $\Delta'$, its affine span $\Aff(\sembunpipj)$ satisfies the following conditions:

* It does not contain other elements of $\Delta'$.

* It does not contain a point $p_l$ with $l>i, j$.

Then, there exists $g'\in k[x^{\pm1}, y^{\pm1}]$ such that $\trop(g')=\trop(g)$ and

\[
\trop(V(f, g'))|_{\cSI' \cup \cPI}=D|_{\cSI' \cup \cPI}.
\]

Let $g=\sum_{i, j}d_{ij}x^iy^j$ and $\mathcal{C}$ the union of the elements of $\Delta'$.
We set $\tilde{d}_{\mathbf{i}}=d_{\mathbf{i}}$ for all $\mathbf{i}\in \mathbb{Z}^2\setminus \{p_1, \dots, p_n\}$.
We will construct $g'=\sum_{\mathbf{i}}\tilde{d}_{\mathbf{i}}\mathbf{x}^{\mathbf{i}}$ by determining the coefficients $\tilde{d}_{p_{i}}$ ($i=1, \dots, n$) inductively.
First, we set $\tilde{d}_{p_1}=d_{p_1}$.
Assume that we determine the coefficients $\tilde{d}_{p_i}$ ($1, \dots, t-1$).
Let $g_{t-1}=g-\sum_{i=1}^{t-1}d_{p_i}\mathbf{x}^{p_i}+\sum_{i=1}^{t-1}\tilde{d}_{p_i}\mathbf{x}^{p_i}$, $T$ be the component of $\mathcal{C}$ containing $p_t$, and $m=\min \{i\in \mathbb{Z}\ |\ p_i\in T\}$.
Note that the restriction of the ordering $p_1< \dots < p_n$ to $T$ is normal by the assumption.
If $t=m$, we set $d'_{p_t}=d_{p_t}$.
If $t>m$, the unique path $p_mTp_t$ contains the element $\sembunpspt$ of $\Delta'$.
Note that $\Aff(\sembunpspt)$ does not contain the points $p_{i}$ ($i>t$) by the assumption.
Let $L_{st}\in \cSI'$ be the ray or the line segment corresponding to $\sembunpspt$.
By Corollary <ref>, we determine an element $\tilde{d}_{p_t}\in k$.
In this case, we set $g_t=g_{t-1}-d_{p_t}\mathbf{x}^{p_t}+\tilde{d}_{p_t}\mathbf{x}^{p_t}$, then we have $\val(\tilde{d}_{p_t})=\val(d_{p_t})$ and $\trop(V(f, g_t))|_{L_{st}}=D|_{L_{st}}$.
Note that we can change the coefficient $d_{p_t}$ by Remark <ref>.

By repeating this process, we get the Laurent polynomial $g'=g-\sum_{i=1}^{n}d_{p_i}\mathbf{x}^{p_i}+\sum_{i=1}^{n}\tilde{d}_{p_i}\mathbf{x}^{p_i}$.
Corollary <ref> shows that for all $L\in \cSI'$, we have

\[
\trop(V(f, g'))|_L=D|_L.
\]

Since we have $\trop(g')=\trop(g)$, we have $\cPI(f, g)=\cPI(f, g')\subset \trop(V(f, g'))$ with the multiplicities taken into account by Theorem <ref>.
Thus, we conclude the proof of Theorem <ref>.

Let $f$ and $g$ be two Laurent polynomials in $k[x^{\pm1}, y^{\pm1}]$.
Assume that

\[
\tau(f; \mathbf{i}; P)\geq (resp. >)\ \tau(g; \mathbf{j}; P)
\]

for a point $P\in \mathbb{R}^2$.
Let $\mathbf{v}\in \mathbb{R}^2$ be a vector which is orthogonal to $\mathbf{j}-\mathbf{i}$.
Then, for all $t\in \mathbb{R}$, we have

\begin{eqnarray*}
\tau(f; \mathbf{i}; P+t\mathbf{v})\geq (resp. >)\ \tau(g; \mathbf{j}; P+t\mathbf{v}).
\end{eqnarray*}

This can be shown by direct calculations.

Let $f, g\in k[x^{\pm1}, y^{\pm1}]$ be Laurent polynomials such that $\trop(f)=x\oplus y\oplus 0$ and $\Trop(V(g))$ is smooth, and $D$ a divisor on $\Trop(V(f))$ satisfying $(*)$ and $(\psi)=D-E$.
Assume that the origin is not a vertex of $\Trop(V(g))$ and that each ray of the intersection $\Trop(V(f)\cap \Trop(V(g))$ has exactly one point of $D$.
Then, there exists a Laurent polynomial $g'\in k[x^{\pm1}, y^{\pm1}]$ such that $\trop(g')=\trop(g)$ and $\trop(V(f, g'))=D$.

First, we show that all the connected components of $\Trop(V(f)) \cap \Trop(V(g))$ are in $\cSIfg\cup \cPI(f, g)$.
Let $A$ be a  connected component of $(\Trop(V(f)) \cap \Trop(V(g))) \setminus \cPI(f, g)$.
Since the origin is not a vertex of $\Trop(V(g))$, it is clear that $A$ is either a ray or a line segment.
If the origin is an endpoind of $A$, then it is the intersection of an edge of $\Trop(V(f))$ and the interior of an edge of $\Trop(V(g))$.
Thus, in this case, the origin is the intersection of a smooth vertex of $\Trop(V(f))$ and the interior of an edge of $\Trop(V(g))$.
A vertex $P\neq \mathbf{0}$ of $A$ is the intersection of the interior of an edge of $\Trop(V(f))$ and a smooth vertex of $\Trop(V(g))$.
Therefore, all the multiplicities of the endpoints of $A$ are $1$.
Since $\Trop(V(g))$ is smooth, it is clear that the interior of $A$ does not contain a vertex of $\Trop(g)$.
Therefore, we have $A\in \cSIfg$.

It is sufficient to show that the map $\Phi_2$ is injective, the union of $\Phi_2(\cSI(f, g))$ is a forest and each tree satisfies the conditions in Theorem <ref>.
Let $p_1,\dots, p_n$ be the $0$-dimensional cells of $\Delta_g$.
First, $\Sone(\Trop(V(f)))$ consists of three rays and they have different slopes.
Hence, it is clear that the map $\Phi_2$ is injective.
Next, we show that the union of $\Phi_2(\cSI(f, g))$ is a forest.
Assume that the union of $\Phi_2(\cSI(f, g))$ is not a forest, i.e., it contains a cycle $C$.
Let $p_{i_1}, \dots, p_{i_m}$ be the vertices of $C$ such that $\sembunpijpij\in \Phi_2(\cSI(f, g))$ for all $j=1, \dots,m$ (we regard $p_{i_{m+1}}=p_{i_1}$).

\begin{eqnarray*}
D_x&:=&\{(x, y)\in \mathbb{R}^2\ |\ x>y, 0\},\\
D_y&:=&\{(x, y)\in \mathbb{R}^2\ |\ y>x, 0\},\\
D_0&:=&\{(x, y)\in \mathbb{R}^2\ |\ 0>x, y\}.
\end{eqnarray*}

Let $D_{i_j}$ $(j=1, \dots, m$) be the closures of the domains of $\mathbb{R}^2\setminus \Trop(V(g))$ corresponding to $p_{i_j}$.
Then, $D_{i_j}$ are convex polyhedral sets, and hence, each intersection $D_{i_j}\cap D_{i_{j+1}}$ is contained in exactly one of the edges of $\Trop(V(f))$.
Assume that $D_{i_j}\cap D_{i_{j+1}}$ is contained in a ray $Y_-:=\{(0, y)\in \mathbb{R}^2\ |\ y\leq 0\}$.
Then, we have $D_{i_j}\cap D_0\neq \emptyset$ or $D_{i_{j+1}}\cap D_0\neq \emptyset$.
Therefore, assume that $D_{i_j}\cap D_0\neq \emptyset$.
Then, we have $D_{i_j}\cap D_x= \emptyset$, $D_{i_{j+1}}\cap D_x\neq \emptyset$ and $D_{i_{j+1}}\cap D_0= \emptyset$.
Here, since $D_{i_{j+1}}$ is a convex set and intersects $D_x$, the intersection $D_{i_{j+1}}\cap D_{i_{j+2}}$ must be contained in the ray $XY:=\{(x, y)\in \mathbb{R}^2\ |\ x=y\geq 0\}$.
By similar arguments, we have $D_{i_{j-1}}\cap D_{i_{j}}\subset X_-:=\{(x, 0)\in \mathbb{R}^2\ |\ x\leq 0\}$, and so on.
Thus, if $\sembunpijpij\in \Phi_2(\cSI(f, g))$ is the bold line segment in (a) of Figure <ref>, $\Phi_2(\cSI(f, g))$ must contain the bold line segments in (b) and (c) of Figure <ref>, and hence the bold line segments in (d) of Figure <ref>.
Here, the $2$-dimensional cell of $\Delta_g$ enclosed by the bold line segments in (d) of Figure <ref> corresponds to a vertex of $\Trop(V(g))$.
It must be the origin, and this contradicts the assumption.
Thus, the union of $\Phi_2(\cSI(f, g))$ is a forest.

Finally, we show that each tree satisfies the conditions in Theorem <ref>.
Let $g=\sum_{i, j}d_{ij}x^iy^j$ and $\sembunx \in \Phi_2(\cSI(f, g))$.
Assume that $\sembuny \in \Phi_2(\cSI(f, g))$ ($a_1\neq a_2$).
Then, there exist $P_1, P_2\in Y_-$ such that

\begin{eqnarray*}
\tau(g; (a_1, b); P_1)=\tau(g; (a_1+1, b); P_1)>\tau(g; (i, j); P_1)\ (\forall (i, j)\neq (a_1, b), (a_1+1, b)),\\
\tau(g; (a_2, b); P_2)=\tau(g; (a_2+1, b); P_2)>\tau(g; (i, j); P_2)\ (\forall (i, j)\neq (a_2, b), (a_2+1,b)).
\end{eqnarray*}

This contradicts to Lemma <ref>.
Thus, $\Aff(\sembunx)$ does not contain other elements of $\Phi_2(\cSI(f, g))$.
The same argument shows that if $\sembunij\in \Phi_2(\cSI(f, g))$, then $\Aff(\sembunij)$ does not contain other elements of $\Phi_2(\cSI(f, g))$.
Hence, each tree satisfies the conditions in Theorem <ref>.

(L1) at (0,0);
(L2) at (2,0);
(L3) at (2.5,0);
(L4) at (4.5,0);
(L5) at (5,0);
(L6) at (7,0);
(L7) at (7.5,0);
(L8) at (9.5,0);
(L9) at (0,2);
(L10) at (2.5,2);
(L11) at (5,2);
(L12) at (7.5,2);
(L13) at (0.5,0.4);
(L14) at (0.9,0.4);
(L15) at (3,0.4);
(L16) at (3.4,0.4);
(L17) at (3,0.8);
(L18) at (5.5,0.4);
(L19) at (5.9,0.4);
(L20) at (5.5,0.8);
(L21) at (8,0.4);
(L22) at (8.4,0.4);
(L23) at (8,0.8);

[very thick](L13)–(L14);
[very thick](L15)–(L16)–(L17);
[very thick](L20)–(L18)–(L19);
[very thick](L21)–(L22)–(L23)–cycle;

[label=below:(a)] (a) at (1,-0.25);
[label=below:(b)] (b) at (3.5,-0.25);
[label=below:(c)] (c) at (6,-0.25);
[label=below:(d)] (d) at (8.5,-0.25);

Elements of $\Phi_2(\cSI(f, g))$.

§ EXAMPLES

In the following, let $k=\mathbb{C}\{\!\{t\}\!\}$ be the field of Puiseux series with coefficients in the complex numbers.

\begin{eqnarray*}
&&f=t^3x^2y^2+t^2x^2y+t^2xy^2+xy+x+y+t\in k[x^{\pm 1}, y^{\pm 1}],\\
&&g=t^3x^2y^2+xy+x+y\in k[x^{\pm 1}, y^{\pm 1}].
\end{eqnarray*}

Then, the tropical curves $\Trop(V(f))$ and $\Trop(V(g))$ are as in Figure <ref>, and hence the intersection $\Trop(V(f))\cap \Trop(V(g))$ is the union of the elements of $\cLS(f, g)$.
If we set $\cSI'=\cLSfg$, then it is acyclic with respect to $\Phi_2$.

(L1) at (-5.5,-1);
(L2) at (-5,-1);
(L3) at (-5.5,-0.5);
(L4) at (-5,-0.5);
(L5) at (-4.5,-0.5);
(L6) at (-5,0);
(L7) at (-4.5,0);

(L8) at (-3,-1);
(L9) at (-3.5,-0.5);
(L10) at (-3,-0.5);
(L11) at (-2.5,0);

[very thick] (L2)–(L4);
[very thick] (L3)–(L4);
[very thick] (L4)–(L7);

[very thick] (L10)–(L8);
[very thick] (L10)–(L9);
[very thick] (L10)–(L11);

(A1) at (-0.5,-1);
(A2) at (1,-1);
(A3) at (-0.5,0.5);
(A4) at (0,-1);
(A5) at (0.5,-1);
(A6) at (-0.5,-0.5);
(A7) at (0.5,-0.5);
(A8) at (-0.5,0);
(A9) at (0,0);

[decorate, decoration=snake, amplitude=0.5pt, segment length=2.5pt] (A3)–(A1);
[decorate, decoration=snake, amplitude=0.5pt, segment length=2.5pt] (A1)–(A2);
[decorate, decoration=snake, amplitude=0.5pt, segment length=2.5pt] (A2)–(A3);
[very thick] (A6)–(A8);
[very thick] (A4)–(A5);
[very thick] (A7)–(A9);

[decorate, decoration=snake, amplitude=0.5pt, segment length=2.5pt] (A1)–(-1,-1.5);
[decorate, decoration=snake, amplitude=0.5pt, segment length=2.5pt] (A2)–(1.5,-1.25);
[decorate, decoration=snake, amplitude=0.5pt, segment length=2.5pt] (A3)–(-0.75,1);

[label=below:$\Delta_f$] (a) at (-4.9,-1.75);
[label=below:$\Delta_g$] (b) at (-3,-1.75);
[label=below:$\Trop(V(f))$ and $\Trop(V(g))$] (c) at (0.25,-1.75);

The tropical curves and dual subdivisions in Example <ref>.

Here, we ask: why do we need the acyclicity condition that the map $\Phi_2|_{\cSI'}$ is injective and the union of the elements of $\Phi_2(\cSI')$ is a forest?
There are two examples which explain this question.

\begin{eqnarray*}
&&f=xy^3+t^2xy^2+y^3+t^5xy+ty^2+t^5y+t^{10}\in k[x^{\pm 1}, y^{\pm 1}],\\
&&g=ax+by+1\in k[x^{\pm 1}, y^{\pm 1}]\ (\val(a)=\val(b)=0).
\end{eqnarray*}

Then, the tropical curves $\Trop(V(f))$ and $\Trop(V(g))$ are as in Figure <ref>, and hence the intersection $\Trop(V(f))\cap \Trop(V(g))$ is the union of the elements of $\cLS(f, g)$, and the stable intersection divisor is

\[
E=(0, 0)+(0, -1)+(0, -4)+(0, -5).
\]

\[
D=\left(0, -\frac{1}{4}\right)+\left(0, -\frac{3}{4}\right)+\left(0, -\frac{13}{3}\right)+\left(0, -\frac{14}{3}\right).
\]

Then, it is clear that there exists a tropical rational function $\psi$ on $\Trop(V(f))$ satisfying $\Supp(\psi)\subset \Trop(V(f))\cap \Trop(V(g))$ and $(\psi)=D-E$.
Let $L_1=\overline{(0, 0)(0, -1)}$, $L_2=\overline{(0, -4)(0, -5)}$ and $\cSI'=\cLS(f, g)=\{L_1, L_2\}$.
Note that the map $\Phi_2|_{\cSI'}$ is not injection.
Assume that $\trop(V(f, g))|_{\cSI'}=D$.

First, we consider $\trop(V(f, g))|_{L_1}$.
We set

\[
g-\frac{a}{t^5y}f=:\sum_{i, j}e_{ij}x^iy^j.
\]

Then, we have $e_{00}=1-a$.
Then, by Proposition <ref>, we have $\val(1-a)=1/4$.

Next, we consider $\trop(V(f, g))|_{L_2}$.
We set

\[
g-\frac{a}{y^3}f=:\sum_{i, j}e'_{ij}x^iy^j.
\]

Then, we have $e'_{00}=1-a$.
Then, by Proposition <ref>, we have $\val(1-a)=1/3$.
This is a contradiction.
Therefore,  there does not exist $g'\in k[x^{\pm1}, y^{\pm1}]$ such that $\trop(g')=\trop(x+y+1)$ and $\trop(V(f, g))|_{\cSI'}=D$.

Example <ref> explains why we need the assumption that the map $\Phi_2|_{\cSI'}$ is injective.

(L1) at (-4.5,-1);
(L2) at (-4.5,-0.5);
(L3) at (-4.5,0);
(L4) at (-4.5,0.5);
(L5) at (-4,0.5);
(L6) at (-4,0);
(L7) at (-4,-0.5);

(L9) at (-3,-1);
(L10) at (-2.5,-1);
(L11) at (-3,-0.5);

[very thick] (L4)–(L5);
[very thick] (L2)–(L7);

[very thick] (L9) to [out=18,in=162] (L10);
[very thick] (L9) to [out=-18,in=198] (L10);

(A1) at (0,1);
(A2) at (0.5,0.5);
(A3) at (0.5,0);
(A4) at (0,-0.5);
(A5) at (0,-1);

[very thick] (A1)–(0,1.5);
[very thick] (A4)–(A5);

[decorate, decoration=snake, amplitude=0.5pt, segment length=2.5pt] (0,1.5)–(-1.5,1.5);
[decorate, decoration=snake, amplitude=0.5pt, segment length=2.5pt] (0,1.5)–(0,-1.5);
[decorate, decoration=snake, amplitude=0.5pt, segment length=2.5pt] (0,1.5)–(1,2.5);

[label=below:$\Delta_f$] (a) at (-4.2,-1.75);
[label=below:$\Delta_g$] (b) at (-2.7,-1.75);
[label=below:$\Trop(V(f))$ and $\Trop(V(g))$] (c) at (0.5,-1.75);

The tropical curves and dual subdivisions in Example <ref>.

If we regard the two bold line segments in $\Delta_g$ as different things as in Figure <ref>, they form a cycle.
Thus, we can regard the assumption that the map $\Phi_2|_{\cSI'}$ is injective is a part of the assumption that the union of the elements of $\Phi_2(\cSI')$ is a forest.

\begin{eqnarray*}
&&f=t^3x^3y^3+tx^3y^2+tx^2y^3+x^2y^2+tx^2y+txy^2+txy+t^3\in k[x^{\pm 1}, y^{\pm 1}],\\
&&g=ax+by+1\in k[x^{\pm 1}, y^{\pm 1}]\ (\val(a)=\val(b)=0).
\end{eqnarray*}

Then, the tropical curves $\Trop(V(f))$ and $\Trop(V(g))$ are as in Figure <ref>, and hence the intersection $\Trop(V(f))\cap \Trop(V(g))$ is the union of the elements of $\cLS(f, g)$, and the stable intersection divisor is

\[
E=(-2, 0)+(-1, 0)+(0, -2)+(0, -1)+(1, 1)+(2, 2).
\]

\[
D=\left(-\frac{7}{4}, 0\right)+\left(-\frac{5}{4}, 0\right)+\left(0, -\frac{5}{3}\right)+\left(0, -\frac{4}{3}\right)+\left(\frac{4}{3}, \frac{4}{3}\right)+\left(\frac{5}{3}, \frac{5}{3}\right).
\]

Then, It is clear that there exists a tropical rational function $\psi$ on $\Trop(V(f))$ satisfying $\Supp(\psi)\subset \Trop(V(f))\cap \Trop(V(g))$ and $(\psi)=D-E$.
Let $L_1=\overline{(1, 1)(2, 2)}$, $L_2=\overline{(-1, 0)(-2, 0)}$, $L_3=\overline{(0, -1)(0, -2)}$ and $\cSI'=\cLS(f, g)=\{L_1, L_2, L_3\}$.
Note that the union of the elements of $\Phi_2(\cLS')$ is not a forest.
Assume that $\trop(V(f, g))|_{\cLS'}=D$.

First, we consider $\trop(V(f, g))|_{L_1}$.
We set

\[
g-\frac{a}{tx^2y^2}f=:\sum_{i, j}e_{ij}x^iy^j.
\]

Then, we have $e_{01}=b-a$.
Then, by Proposition <ref>, we have $\val(b-a)=1/3$.

Next, we consider $\trop(V(f, g))|_{L_2}$.
We set

\[
g-\frac{b}{txy}f=:\sum_{i, j}e'_{ij}x^iy^j.
\]

Then, we have $e'_{00}=1-b$.
Then, by Proposition <ref>, we have $\val(1-b)=1/4$.

Finally, we consider $\trop(V(f, g))|_{L_3}$.
We set

\[
g-\frac{a}{txy}f=:\sum_{i, j}e''_{ij}x^iy^j.
\]

Then, we have $e''_{00}=1-a$.
Then, by Proposition <ref>, we have $\val(1-a)=1/3$.

Thus, we have

\begin{eqnarray*}
\end{eqnarray*}

However, we also have

\[
\val(1-a)=\val((1-b)+(b-a))=\frac{1}{4}.
\]

This is a contradiction.
Therefore,  there does not exist $g'\in k[x^{\pm1}, y^{\pm1}]$ such that $\trop(g')=\trop(x+y+1)$ and $\trop(V(f, g))|_{\cSI'}=D$.

Example <ref> explains why we need the assumption that the union of the elements of $\Phi_2(\cSI')$ is a forest.

(L1) at (-5.5,-1);
(L2) at (-5,-0.5);
(L3) at (-4.5,-0.5);
(L4) at (-5,0);
(L5) at (-4.5,0);
(L6) at (-4,0);
(L7) at (-4.5,0.5);
(L8) at (-4,0.5);
(L9) at (-3,-1);
(L10) at (-2.5,-1);
(L11) at (-3,-0.5);

[very thick] (L6)–(L7);
[very thick] (L2)–(L3);
[very thick] (L2)–(L4);
[very thick](L9)–(L10)–(L11)–cycle;

(A1) at (-1,0);
(A2) at (0,-1);
(A3) at (-0.5,0);
(A4) at (0,-0.5);
(A5) at (-0.5,0.5);
(A6) at (0.5,0.5);
(A7) at (0.5,-0.5);
(A8) at (1,1);

[very thick] (A6)–(A8);
[very thick] (A1)–(A3);
[very thick] (A2)–(A4);

[decorate, decoration=snake, amplitude=0.5pt, segment length=2.5pt] (0,0)–(-1.5,0);
[decorate, decoration=snake, amplitude=0.5pt, segment length=2.5pt] (0,0)–(0,-1.5);
[decorate, decoration=snake, amplitude=0.5pt, segment length=2.5pt] (0,0)–(1.5,1.5);

[label=below:$\Delta_f$] (a) at (-4.5,-1.75);
[label=below:$\Delta_g$] (b) at (-2.7,-1.75);
[label=below:$\Trop(V(f))$ and $\Trop(V(g))$] (c) at (0.5,-1.75);

The tropical curves and dual subdivisions in Example <ref>.

Directions for future research are as follows.
We have explained why we need the acyclicity condition.

However, we do not know whether we need the assumption about an order of the endpoints of the elements of $\Delta'$.

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