数理解析研究所講究録
第 1764 巻 2011 年 149-164

149

Basic Topics on Tropical Geometry andSingula.rities( トロピカル幾何と特異性に関する基本的トピックス )Goo Ishikawa
Department of Mathematics, I-Iokkaido University, Japan( 石川 剛郎 ，北海道大学数学教室，日本)

e-mail: ishikawa@math.sci.hokudai.ac.jp
Motivated from real algebraic geometry, Viro, Mikhalkin, Shustin, Itenberg, and other mathematicians have developed the ”tropical (algebraic)
geometry” [8]. Algebraic curves are tropicalized to piecewise-linear curves.

The method was used to construct topological types of real algebraic curves
in Hilbert’s 16th problem [24].
In this rough sketch, we present several basic topics of tropical geometry, in particular, the notion of hyperfields introduced by Viro recently[25].

[ Tropical Limits of Operations ]Let $R+$ denote the set of non-negative real numbers.
We fix $h>0$ and consider the bijection$hlog:R_{+}-RU\{-\infty\}$defined by$urightarrow h\log u$,$e^{\grave{7_{l}}\tau-\chi}$

.

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On

{-00}, two operations$R\cup$$\{\begin{array}{l}x+hy;=h\log(e^{X}\Gamma\iota+e\mathscr{N}_{l})x\cross hy;=l\iota\log(e^{X}\cdot e^{y})=x+y\end{array}$are induced from the summension and the multiplication on$R+\cdot$

Set $m= \max\{x,y\}.$ Then we have.

$h\log(e^{\frac{\prime\prime\prime}{;_{1}}})\leq x+1\iota y\leq h\log(+e)$

namely,,/l $y\leq\uparrow n+h$ log2.

$m\leq x$ 十Therefore we have that$\lim_{h\downarrow 0}(x+l\iota/t)-==\max\{x,y\}$.

[Tropical Semi-Ring]$R_{trop}=R\cup${ oo} with the two operations$-$$\prime\prime x+y’’:=\max\{x,\iota J\}$,$\prime\prime x\cdot y’’:=x+y$,

is called the tropical semi-ring (or the max-plus algebra).
Moreover we set” $\chi/y’’=x-y$ if $y\neq-$ oo. Note that there is no tropicalsubtraction. The tropical sum is idempotent:$\prime\prime x+x’’=x$$-$,oo being the tropical zero.

[Tropical Polynomials]
For a finite subset$A\subset Z^{n}$, consider a “tropical” (Laurent) polynomial$F(x)=^{JJ} \sum_{i\in A}c_{i}x^{j\prime\prime}=\max\{c_{i}+j\cdot x|j\in A\}$$(c_{j}\in R)$$Y_{F}\subset R^{l?}$, which is a PL-function on

is defined by$F$,. Then the tropical hypersurface
as the corner locus of .
$R^{\mathfrak{l}1}$

$F$

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Example 1. (tropical line). We consider$F(x_{1},x_{2})= \prime\prime ax_{1}+bx_{2}-\vdash c’’=\max\{x_{1}+a, x_{2}+b, c\}$Then

$Y_{F}$.

consists of three half-lines meeting at one point.

[Tropical Hyperfields 1
We define a multi-valued addition
we seta$Y$

on$R\cup${-00}: For $a,b\in R\cup$ {-00},$Yb:=\{\begin{array}{l}\max\{a,b\},\{\iota f\in R\cup\{-\infty\}|y\leq a\}, (a=b)(a\neq b)\end{array}$

The multiplication is defined by the ordinary addition.
lNe set $Y=$ $(R\cup \{- 00\}, Y,+)$ and we call it the tropical hyperfield.
This implies the natural definition of ”tropical zero”.
Example 2. For $a\in R$, we define the function $xYa:Yarrow$ Y. Then we havegraph $(xYa)=\{y=a,x<a\}\cup\{y=x,a<x\}\cup\{y\leq a,x=a\}$ .

152[Definition of Hyperfields ]Suppose there are given, on a set X, a multi-valued binary operation anda single-valued binary operation .
Then
is called a hyperfield if$T$$(X, T,\cdot)$$aT(bTc)=(aTb)TC$$aTb=bTa$ ,$\exists 0\in X,$$0_{T}a=a$ , for$\forall a\in X,\exists\iota-a\in X$

any a$\in X$(minus a)$s$,uch that $0\in aT(-a)$ .

. $c\in aTb=(-c)\in(-a)_{T}(-b)$

The operation . is commutative, associative and0. $a=0$ holds for any $a\in X$ ,$(X\backslash \{0\}, \cdot)$is a commutative group, which will be denoted bythe ”distributive law” holds:$a\cdot(bTc)=(a\cdot b)_{T}(a\cdot c)$ ,$(bTc)\cdot a=(b\cdot a)_{T}(c\cdot a)$.

Lemma 3. The tropical hyperfield $Y=(R\cup\{-\infty\},Y,+)$ is a $hype\phi eld$ .

In fact we have

The zero-element is-oo.
For $a\in Y,$ $-a$ equals$a,$$since-\infty\in aYb\Leftrightarrow b=a$.

The commutative group $Y^{\cross}=(R,+)$ , the unit being$0\in$

R.

$X^{\cross}$

,

153

[Tropical Hypersurfaces and Newton Polyhedral
For a tropical Laurent polynomial$F(x)= \prime\prime\sum_{j\in A}c_{\int}x^{j}$”, we define$v=-c:Aarrow R$

by $v(j)$ $:=-c_{i},$ $(j\in A)$ . Then we set$LJ(v)$ $:=$

convex hull$\{(j,\iota J)\in R^{\mathfrak{s}\iota}\cross R|j\in A,y\geqq v(j)\}\subset R^{n+1}$

We set, and A the union of compact faces
convex hullof $U(v)$ . We callthe Newton polyhedra of .
$\Delta=\Delta_{F}=$$(A)\subseteq R^{\mathfrak{l}l}$$F$$\Delta=\Delta_{F}$$\pi(j,y)$ $:=j$ . An integral
Then projects to in bijection bysubdivision of is induced from . We obtain the convex function
having as its graph.
$\tilde{\Delta}$$\pi:\overline{\Delta}arrow R^{n},$$\Delta$$\tilde{\Delta}$$\Delta$$R$$\overline{v}:\Deltaarrow$$\tilde{\Delta}$$\overline{\Delta}$

The tropical hypersurface is an $(n-1)$ -dimensional regular polyhedral complex. (Regularity condition: the boundary of each i-cell is a unionof $(i-1)$ -cells.)
Along each $(n-1)$ -cell , two functions $c_{j}+j\cdot x,$ $c_{k}+k\cdot x$ have thesame value. From $c_{j}+j\cdot x=c_{k}+k\cdot x$ , we have the equation$Y_{F}$$l$$(k-j)x+(c_{k}-c_{f})=0$

of the hyperplane containing . Then the integer vector $k-f$ is orthogonal
to . Then there exist the unique positive integer
and the primitive
integer vector such that $k-j=\tau v_{I}n_{l}$ .
For each $(n-2)$ -cell , and $(n-1)$ -cells
adjacent to , if
we fix a co-orientation of and take primitive orthogonal vectors , thenwe have the balanced condition
$I$

$l$

$w_{I}$$n_{l}$$C$$C$$I_{1},l_{2},\ldots,I_{m}$$C$$n_{I_{j}}$.

$w_{I_{1}}n_{I_{1}}+w_{l_{2}}n_{I_{2}}+\cdots+w_{I_{ll}},n_{I_{1l}},=0$

154

Thus the tropical hypersurface is an $(n-1)$ -dimensional weighted
rational polyhedral complex satisfying the regularity condition and thebalanced condition.
is invariant under the deformations, called
Tropical hypersurface
the fundamental deformations, of the tropical Laurent polynomial .
(1) Replace by : $Aarrow R$, c’ $(j)=c(j)+$ const..
(2) Replace
by $A’=A+j_{0},j_{0}\in Z$“ and by : $A’arrow R,$ $c’(j+j_{0})=$$c(f)$ .
(3) Replace $c:Aarrow R$ by :such that convex hull $A’=\Delta$ and.
the convex function$Y$$Y_{F}$

$F$

$c$

$c’$

$A$

$c’$

$c$

$c’$

$A^{l}arrow R$$\overline{-c’}=\overline{-c}$

K Legendre Transformations]
Consider the contact manifold$M=R^{2\prime\iota+1}$

with coordinates$(x,y,p)=(x_{1}, ...,x_{1l},y,pl, ...,p_{n})$and with the contact form
Note $that-\theta=d(\Sigma_{i=1}^{l?}1$
Legendrian fibration:$\theta=d\iota J-\Sigma_{i=1}^{ll}p_{i}dx_{i}$.

. Then we have the double$R^{l1+1}R^{2\prime l+1}\underline{7t_{1}}arrow\pi_{2}R^{n+1}$$\pi_{1}(x,y,p)=(x,y),$$\pi_{2}(x,\iota J,p)=(\tilde{y},p)$,,$\tilde{y}=\Sigma_{i=1}^{n}p_{i}x_{i}-y$.

, the Legendre transforFor a function $h:\Deltaarrow R$ on a convex set
mation of is defined as the set of supporting hyperplanes of the epi-graph of$\Delta\subset R^{n}$

$h$

$h$

.

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Lemma 4. The graph of tropical polynomial function$F(x)=^{JJ} \sum_{i\in A}c_{i}x^{j\prime\prime}$

and the graph of the convex function
tions to each other.

$\overline{-c}:R^{n}arrow R$

are the Legendre transforma-$\backslash$$\cdot$$rightarrow$$\}l$$\sim$$|$$\sqrt{}$:$-^{:::}::$
We consider the topological classification problem of tropical polynomial functions preserving corner loci.
Definition 5. Two tropical polynomials $F(x)$ and $G(x)$ are called topologically equivalent if there exist homeomorphisms $\Phi:R^{n}arrow R^{n}$ and $\Psi:Rarrow R$
such that$\Psi(F(x))--\cdot G(\Phi(x)),$ $\Phi(Y_{F})=Y_{G}$ .

Proposition 6. There exis a sein ialgebraic setof codim $>0$ such that,, the decomposition of is simplicial.
for any, the family $F_{c}(x),c\in U$ of tropicalFor each connected component ofpolynomial functions is topologically trivial.
$\Sigma\subset R^{A}$$ts$$c\in R^{A}\backslash \Sigma$$\Delta$$U$$R^{A}\backslash \Sigma$[Topological Bifurcations of Singularities]
The topology of a tropical polynomial with a non-simplicial decomposition bifurcates into a generic tropical polynomial.
Example 7. Let us consider the tropical polynomial$F=”$ $0+0x_{1}+0x_{2} \cdot-\vdash 0x_{1}x_{2’’}=\max\{0,x_{1},x_{2},x_{1}+x_{2}\}$.

156
Then has the deformation:$F$$F_{\lambda}=\prime\prime\lambda+0x_{1}+0\chi_{2}+0x_{1}x_{2}$$”= \max\{\lambda,x_{1},x_{2},x_{1}+x_{2}\}$The tropical curve
bifurcates into
sition of Newton polyhedronbifurcates into

, $(A \in R,\lambda\neq 0)$ .

$J_{F_{\lambda}}’(\lambda>0,\lambda<0)$$Y_{F}$$\Lambda_{F}$$\Delta_{F_{\lambda}}$. The decompo-$(A>0,\lambda<0)$.

$Y_{F_{\lambda}}$

$0$

$0$

$0$

$0$

$rightarrow$$0$$\lambda=0$$\lambda<0$$0$$\lambda$$0$$rightarrow$

$0$

$0$

$\lambda$$0$$\lambda>0$[Amoeba and Pachworking]For a complex Laurent polynomial$f(z)= \sum_{j\in A}b;z^{\int}$c-:$C[z_{1}^{\pm},\ldots,z_{n}^{\pm}]$,$b_{j}\in C^{\cross}$,

we have a hypersurface$Z_{f}=\{\angle’\in(C^{\cross})^{1l}|f(z)=0\}\subset(C^{\cross})^{n}$in the complex torus.
For a given function $v:A-arrow R$ , consider the family of polynomials,$(C^{\cross})^{t1}$$f_{t}=f_{t^{\overline{t}^{)}}}(z):= \sum_{j\in A}b_{1}t^{-v(j)_{Z}j}$,$(t>0)$.

We call it the patchworking poly nomial induced by and . Note that $f_{1}=f$ .
$f$

$v$

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Let us define Log, : $C”arrow(R\cup\{-\infty\})^{tl}$ by${\rm Log}_{t}(z_{1}, \ldots,z_{l})=(\log_{t}|z_{1}|,\log_{i}|z_{n}|)$We setand we call it the amoeba of$\mathcal{A}_{f}={\rm Log}(Z_{f})\subset R^{tl}$.

$z_{f}$.

Proposition 8. (Viro, Kapranov)$\lim_{farrow\infty}$

Hausdorff-dist$({\rm Log}_{f}(Z_{ft}),Y_{f_{t^{v}rop}})=0$

where$f_{trop}(x):= \sum_{j\in A}(-v(j))x^{j\prime\prime}=_{j\in A}\max(j\cdot x-v(j))$(Legendre transformation of ).
$v$

Example 9. Amoeba of $f(z_{1},z_{2})\cdot 1+1$ .

[Puiseux Series and Non-Archimedean Amoebal
Let us denote by $C[R]$ the group algebra of the additive group over C.
We consider its formal version:
A Puiseux-Laurent series of real power (Hahn series[4]) is given by
$R$

$a$

— $a$
$(s)= \sum_{p\in I}\alpha_{p}s^{p}$where
We set$\alpha_{p}\in C^{\cross}$$C((R))$ $:=$and the support $I=I_{a}\subset R$ of is a well-ordered subset.
$a${ $a(s)|a(s)$ : Puiseux-Laurent series of real power}$\cup\{0\}$.

158
Lemma 10. $K=C((R))$ is an$nlg_{CJ}b\dagger’aicall\iota$

closed field.

Define the valuation val: $C((R))arrow$ RU$\{\infty\}$

on $C((R))$ byval(O)

val $(a)$ $:= \min I_{a}\in R,$ $(a\in C((R))\backslash \{0\})$ ,$=\infty$,

Then we have that val $(a)=\infty$ if and only if $a=0$, and thatval $(ab)=$ val $(a)+$ val $(b)$ ,val $(a+b) \geq\min$ {val $(a)$ , val $(b)$ }.

We define the non-Archimedes norm on $C((R))$ by$\Vert a\Vert:=e^{-va1(a)}$$(a\in C((R))^{\cross})$,$\Vert 0\Vert=0$.

Then we have the tropical triangular inequality$\Vert a+b\Vert\leq\max\{\Vert a\Vert, \Vert b\Vert\}="$$\Vert a\Vert+\Vert b\Vert$‘’

Define $Log:C((R))^{rl}arrow(R\cup\{-\infty\})^{l}$ by${\rm Log}(a_{1\prime\cdots\prime}.a_{\dagger\iota})$$:^{---}$$(\log\Vert a_{1}\Vert,\ldots,\log\Vert a_{n}\Vert)$$–$$(-\cdot va1(a_{1}),\ldots,-va1(a_{n}))$.

Given a Laurent polynomial $f(z)=\Sigma_{i}a_{j}z^{j}\in K[z,z^{-1}]$ , we define$Z_{f}:=\{z\in(K^{\cross})^{\prime\iota}|f(z)=0\}\subset(K^{\cross})^{n}$RU
Its Log-image
amoeba of .
Define a tropical Laurent polynomial$A_{f}:={\rm Log}(Z_{f})\subseteq($$\{-\infty\})^{n}$.

is called the non-Archimedean$z_{f}$$f_{trop}(x)$$:=\prime\prime\Sigma_{j\in A}\log\Vert a_{j}\Vert xf^{\prime\prime\prime\prime}=\Sigma_{j\in A}$$= \max_{j\in\Lambda}(j\cdot x-va1(a_{j}))$(-val$(a_{j})$)$x^{j\prime}$.

We call $f_{trop}(x)$ the tropicalization of $f(z)$ .

Proposition 11. (Kapranov) Non-A rchimedean amoeba is a tropical hypersurface: We have $A_{[}=Y_{f_{trop}}$ .

159[Triangle hyperfield]
On $R+$ , define the multi-valued addition$a\nabla b;=$

$=$

$\{c$

欧 $R\dashv-||a-b|\leq c\leq a+b\}$$\{|z+\cdot w|||z|=a, |w|=b\}$.

This reminds us the superposition of waves.
Then
is a hyperfield.
$R_{+}^{tri}=(R_{+},\nabla, \cdot)$[Amoeba hyperfield ]By the bijection $log:R+arrow R\cup$ { oo}, we have the hyperfield$-$$\log(R_{+}^{tri})-=(R\cup\{-\infty\}, Y’+)$ ,which is called the amoeba hyperfield:

a $Yb:=\{c\in R\cup\{-\infty\}|\log(|e^{0}-e^{b}|)\leq c\leq\log(e^{a}+e^{b})\}$ .

[Tropical Limits of Amoeba Hyperfield]
Define, on $R\cup\{-\infty\}$ ,$a$$Y_{1z}b;=h(\frac{a}{h}Y_{f_{1}}^{b}\cdot)$$=$$\{c\in R(\lrcorner\{-\infty\}|$$h\log(|e^{l1}\gamma, -e^{l?}\tau_{l}|)\leq c\leq h\log(e\tau_{l}+e\pi)ab,$ $\}$$a$$Y_{h}b$

RU {-00} $|-\infty\leq c\leq a+h$ log2$=$$\{c\in$$=$$[-\infty, a]=:aYa$ .

If $a\neq b$ , then a $Y\prime_{2}barrow\{\max\{a,b\}\}$ .
$a$$\varliminf_{?*0}$$Y_{2}b=aYb$ ,$\lim_{/2arrow 0}\log(R_{+}^{tri})_{l\iota}arrow Y$(: tropical hyperfield).

$\}$

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[Complex Tropical hypetfield 1
We define a multi-valued addition on : Let $a,b\in$ C. If $|a|\neq|b|$ , thenwe set $aarrow b:=a$ if $|a|>|b|$ , and $aarrow b:=b$ if $|a|<|b|$ .
Suppose $|a|=|b|$ . If $b\neq-(l$ , then
-

$a-b$

$:=$

$C$

[the shortest arc connecting a andon the circle $\{z\in C||z|=|a|\}]$ .
$b$

If $b=-a$ , then set
$a-b$

$:–=$

We define the $com$plex tropical$\mathcal{T}C:=$( $C,$$arrow$$\{z\in C||z|\leq|a|\}$$h\iota J$

perfield by, the usual multiplication).

On , we consider the bijcction$C$.

$S_{ll}$: $Carrow C$ defined by$S_{h}(z)::\{\begin{array}{ll}|z|^{\frac{1}{\prime_{1}}}\frac{z}{|z|} (z\neq 0),0 (z=0).\end{array}$and we define$z+hw::^{-}=S_{h}^{-1}(S_{h}(z)+S_{h}(w))$.

Then we have a family of fields $(C,+h, \cross),$ $h>0$ .

Theorem 12. (Viro [25]) Let$\Gamma=\{(z,w,z+hw,h)\in C^{3}\cross R+|(z,w,h)\in C^{2}\cross R_{+}\}$

Then$\overline{\Gamma}\cap(C^{3}\cross\{0\})\cdot.\cdot:=\{(a, b, aarrow b, 0)|(a,b)\in C^{2}\}$

.

.

161

[Viro’s Diagram $2010\beta$
Thus we have the diagram:[Real Tropical Hyperfield]$?Question$: What is the real counterpart of the complex tropical hyperfield

We are naturally led to define the multi-valued addition, we set
duced from -on : For$C$$a,b\epsilon:$.

$arrow R$$R$$\{\begin{array}{ll}a-Rb := a if a|>|b|,a-Rb := b if |a|<|b|,a\sim 1\mathfrak{i}a ;=a,a-n(-0) ;= [-a,a].\end{array}$$\infty\inftyarrowrightarrow$
$0ba$

$b0$

Theorem 13. $(R,-R, \cross)$ is a$h|J$$\mathfrak{g}$$0a=b-\alpha$$0$$\mathfrak{g}$perfield. Moreover let$\Gamma_{R}=\{(a,b,a+hb,h)\in R^{3}\cross R+|(a,b,h)\in R^{2}\cross R_{+}\}$

Then we have$\overline{\Gamma_{R}}\cap(R^{3}\cross\{0\})\cdot-\cdot=\{(a, b, aarrow Rb, 0)|(a,b)\in R^{2}\}$

.

.

on

$R$

in-

162
The real tropical hyperfie]d is, in some sense, a ”double covering” of. Therefore, “real tropical geometry”
the tropical hyperfield via
can be constructed as a ”double covering” of tropical geometry.
$x\vdash\rangle\log|x|$

K Several Questions 1
Question: Is the complex tropical hyperfield
is algebraically closed, in
an appropriate sense ?
Question: Are the real tropi $(a1$ hyperfield and the tropical hyperfield $Y$
real closed ?
Question: What is the real tropical algebraic geometry ?
$\mathcal{T}C$In Amoeba geometry, it is known the Ronkin function$N_{f}(x):= \frac{1}{(2\pi\sqrt{-}}1^{\cdot}\overline{)}^{\prime l}-\int_{{\rm Log}^{-1}(x)}\log|f(z)|\frac{dz_{1}}{z_{1}}\cdots\frac{dz_{n}}{z_{n}}$. We have $gradN_{f}$ :is linear on each connected component ofand $gradN_{f}$ separates every connected components of$R^{n}\backslash \mathcal{A}_{f}$$R^{n}\backslash$$R^{n}\backslash$$\mathcal{A}_{f}^{f}\mathcal{A}.arrow\Delta\cap Z^{n}$

Question: Can the Ronkin function be described in terms of the amoeba
hyperfield ?

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