next up previous
Next: About this document ...

    

鐃緒申鐃緒申鐃� II 鐃庶験鐃緒申鐃緒申

鐃緒申鐃緒申 1.1-1.4 鐃塾わ申鐃緒申鐃緒申鐃緒申2鐃縦わ申鐃緒申鐃俊￥申鐃薯きなわ申鐃緒申鐃緒申

鐃緒申鐃緒申 1.1   鐃緒申鐃塾各￥申鐃塾常申鐃祝つわ申鐃銃￥申 $R$ 鐃塾ワ申鐃叔ワ申鐃緒申 $I$ 鐃緒申鐃出縁申鐃緒申鐃緒申鐃緒申鐃重�鐃緒申鐃緒申 $V(I)$ 鐃塾鰹申鐃緒申鐃緒申鐃緒申鐃緒申鐃淑わ申鐃緒申鐃緒申

1.
$R=\mbox{${\Bbb R}$ }[X]$ 鐃緒申 $I=(X^5-X)$ 鐃塾常申隋�
2.
$R=\mbox{${\Bbb R}$ }[X,Y]$ 鐃緒申 $I=(X-3,X^2Y-X-42)$ 鐃塾常申隋�
3.
$R=\mbox{${\Bbb R}$ }[X,Y]$ 鐃緒申 $I=((Y-X)^2-(X+Y)^2-(X+Y)^3)$ 鐃塾常申隋�
4.
$R=\mbox{${\Bbb R}$ }[X,Y,Z]$ 鐃緒申 $I=(Y-Z)$ 鐃塾常申隋�

(鐃緒申鐃緒申)

(1) $X^5-X=0$ 鐃緒申 $\mbox{${\Bbb R}$ }$ 鐃叔の駕申鐃緒申鐃緒申里鐃緒申鐃緒申鐃緒申砲覆襦�

\begin{displaymath}X^5-X=X(X^4-1)=X(X^2-1)(X^2+1)=X(X+1)(X-1)(X^2+1) \end{displaymath}

鐃緒申鐃緒申鐃初、 $\{-1,1,0\}$ 鐃緒申多鐃縦常申鐃緒申鐃祝プワ申鐃獣トわ申鐃緒申个茲わ申鐃�

(2) 連立鐃緒申鐃緒申鐃緒申
\begin{align*}& X-3=0 \\ & X^2Y-X-42=0 \end{align*}
鐃塾駕申鐃緒申鐃緒申个茲わ申鐃� 鐃緒申鐃緒申鐃緒申 $(3,5)$ 鐃緒申平鐃縮常申縫廛鐃緒申奪箸鐃緒申鐃緒申鐃塾でわ申鐃暑。

(3) 鐃緒申鐃緒申鐃緒申鐃緒申呂鐃緒申鐃緒申颪靴鐃緒申辰鐃緒申鐃緒申鐃緒申里鐃淑わ申鐃緒申 $Y-X=T, X+Y=U$ 鐃夙わ申鐃緒申鐃夙￥申

\begin{displaymath}T^2-U^2-U^3=0 \end{displaymath}

鐃淑わ申鐃緒申鐃緒申鐃緒申鐃塾駈申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申砲覆襦�(鐃縦まわ申鐃緒申鐃緒申鐃淑削申標鐃術器申鐃緒申 鐃緒申鐃緒申鐃緒け鐃緒申鐃緒申

$T,U$ 鐃祝関わ申鐃緒申鐃塾種申鐃熟講居申鐃叔逸申鐃獣わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申呂鐃緒申任鐃緒申襦� $(T,U)$ 鐃緒申鐃緒申 $(X,Y)$ 鐃緒申鐃潤す鐃祝は￥申単鐃祝わ申鐃緒申鐃� $\pi/4$ 鐃緒申鐃緒申鐃緒申転鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃� $\sqrt{2}$ 鐃旬に縮常申鐃緒申鐃緒申个茲わ申鐃�

\includegraphics[scale=0.5]{113.ps}

鐃緒申鐃緒申鐃緒申鵝◆鐃緒申鐃緒申鐃祝￥申廚如鐃�$X,Y$ 鐃緒申鐃順き鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃銃逸申鐃緒申鐃緒申鐃緒申鐃緒申鐃重�鐃緒申判鐃叔わ申鐃緒申 鐃緒申鐃緒申佞鐃緒申鐃緒申鐃緒申討鐃処い鐃緒申

鐃緒申鐃緒申 1.2   $A=\mbox{${\Bbb R}$ }[X,Y,Z]$ 鐃塾ワ申鐃叔ワ申鐃緒申 $I=(X^2-Y^2-Z)$ 鐃祝つわ申鐃銃￥申鐃緒申鐃緒申鐃巡い 鐃緒申鐃緒申鐃緒申鐃淑わ申鐃緒申鐃緒申
1.
$V(I)$ 鐃緒申平鐃緒申 $V(Z-1)$ 鐃緒申鐃准っわ申鐃准わ申鐃�(鐃緒申鐃淑わ申鐃� $V(I)\cap V(Z-1)$ 鐃塾図件申鐃緒申鐃緒申鐃緒申鐃淑わ申鐃緒申鐃緒申鐃緒申平鐃縮わ申鐃准っわ申鐃緒け鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃藷グワ申佞鐃� 平鐃縮常申鐃緒申鐃緒申鐃緒申个茲わ申鐃�)

2.
$I$ 鐃緒申 $Z-(X^2-Y^2-Z)$ 鐃夙わ申鐃緒申鐃緒申鐃緒申鐃曙た鐃緒申鐃叔ワ申鐃緒申

\begin{displaymath}J=(X^2-Y^2-Z,Z-(X^2-Y^2-Z)) \end{displaymath}

鐃祝つわ申鐃銃￥申 $A/J$ 鐃緒申 $0$ 鐃叔なわ申鐃緒申 $[f],[g]$ 鐃叔￥申 $[f][g]=0$ 鐃緒申澆鐃緒申鐃緒申鐃塾わ申 鐃緒申鐃夙居申鐃淑わ申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申 $[?]$ 鐃緒申 $?$ 鐃緒申 $A/I$ 鐃叔のワ申鐃初ス鐃緒申表鐃緒申鐃緒申里箸鐃緒申襦�
3.
鐃緒申鐃� $f,g$ 鐃祝つわ申鐃銃￥申

\begin{displaymath}fg=a(X,Y,Z) (X^2-Y^2-Z)+b(X,Y,Z)(Z-(X^2-Y^2-Z)) \end{displaymath}

鐃緒申 鐃淳わ申鐃緒申鐃処う鐃緒申 $a,b \in A$ 鐃緒申尊櫃傍鐃緒申覆鐃緒申鐃緒申鐃�

4.
$A/J$ 鐃塾醐申 $ [Y^4+3Y^2+Z]$ 鐃緒申 $X$ 鐃緒申鐃緒申鐃塾種申 $p(X)$鐃塾ワ申鐃初ス鐃夙わ申鐃緒申表鐃緒申鐃緒申鐃淑わ申鐃緒申鐃緒申鐃縦まり、

\begin{displaymath}[Y^4+3Y^2+Z]=[p(X)] \end{displaymath}

鐃夙なわ申 $p(X)$ 鐃緒申鐃縦居申鐃淑わ申鐃緒申鐃緒申

(鐃緒申鐃緒申)

(1) $X^2-Y^2-1=0.$ 鐃緒申鐃緒申鐃緒申亢鐃緒申鐃緒申任鐃緒申襦�

(2)

\begin{displaymath}J=(X^2-Y^2-Z,Z-(X^2-Y^2-Z))=(X^2-Y^2,Z) \end{displaymath}

鐃夙なって￥申$V(J)$ 鐃緒申平鐃縮常申鐃緒申鐃縦常申鐃� $X^2-Y^2=0$ 鐃夙みなわ申鐃緒申鐃夙わ申鐃叔わ申鐃暑。 鐃緒申鐃緒申鐃熟わ申鐃夙わ申鐃緒申

\begin{displaymath}f=X-Y, g=X+Y \end{displaymath}

(3) 鐃緒申鐃緒申禄鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 $f,g$ 鐃塾種申鐃緒申鐃緒申砲鐃緒申錣縁申鐃緒申鐃緒申鐃� 鐃緒申亮鐃緒申鐃緒申鐃淑わ申弌鐃�
\begin{align*}fg&=X^2-Y^2=(X^2-Y^2-Z)+Z-(X^2-Y^2-Z)+(X^2-Y^2-Z)\\ &=2(X^2-y^2-Z)+(-1)(Z-(X^2-Y^2-Z)) \end{align*}
鐃叔わ申鐃夙わ申鐃緒申 $a=2,b=-1$ 鐃夙わ申鐃緒申个茲わ申鐃�

(4) $A/J$ 鐃塾なわ申鐃叔は￥申 $[X^2]=[Y^2]$, $[Z]=0$ 鐃緒申鐃淑りた鐃獣てわ申鐃緒申錣縁申任鐃緒申襪�鐃初、

\begin{displaymath}[Y^4+3Y^2+Z]=[Y^2]^2+3[Y^2]+[Z]=[X^2]^2+3[X^2]+0=[X^4+3X^2]. \end{displaymath}

鐃醇え鐃緒申 $p=X^4+3X^2$ 鐃夙わ申鐃緒申个茲わ申鐃�

鐃緒申鐃緒申 1.3   $\phi: \mbox{${\Bbb R}$ }[X,Y] \to \mbox{${\Bbb R}$ }[T]$ 鐃緒申

\begin{displaymath}\phi(X)=T^2 ,\qquad \phi(Y)=T(T^2-1) \end{displaymath}

鐃緒申鐃緒申鐃緒申鐃緒申襦ｏ申鐃緒申里箸鐃緒申鐃�
1.
${}^a\phi(0),{}^a\phi(1),{}^a\phi(2),{}^a\phi(-1)$ 鐃薯それぞ鐃緒申鐃緒申覆鐃緒申鐃緒申鐃�
2.
$\ker(\phi)$ 鐃緒申鐃緒申覆鐃緒申鐃緒申鐃� (鐃述ワ申鐃�: $\phi(X)=T^2 ,\qquad \phi(Y)=T(T^2-1)$ 鐃緒申鐃緒申 $T$ 鐃緒申探遒刻申襪鰹申箸鐃� 鐃緒申澆覆鐃緒申鐃緒申鐃�) (鐃緒申鐃緒申縫劵鐃緒申: $\phi(Y)=T(\phi(X)-1)$. 鐃緒申鐃緒申鐃緒申鐃緒申有鐃緒申鐃緒申鐃淑￥申分鐃緒申鐃緒申)鐃緒申弔鐃緒申鐃緒申鐃� $T$ 鐃緒申探鐃叔わ申鐃緒申呂鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申能鐃重�鐃緒申鐃緒申鐃緒申鐃緒申分鐃緒申鐃淑э申辰鐃渋随申犲逸申侶鐃緒申任覆鐃緒申鐃緒申 鐃緒申鐃暑。)

(鐃縮のヒワ申鐃�: $U=T^2, V=T(T^2-1)$ 鐃塾間の関件申鐃緒申鐃緒申鐃緒申鐃祝は￥申 鐃渋わ申3鐃術随申鐃緒申多鐃準式鐃緒申 $\mbox{${\Bbb R}$ }[T,U,V]$ 鐃塾ワ申鐃叔ワ申鐃緒申 $(U-T^2, V-T(T^2-1))$ 鐃塾ワ申鐃緒申屮粉鐃緒申鐃塾わ申鐃緒申鐃緒申$U,V$ 鐃緒申鐃緒申鐃緒申多鐃準式鐃叔わ申鐃緒申鐃塾わ申 鐃緒申鐃出わ申鐃出よい鐃緒申鐃夙わ申鐃塾わ申鐃銃わ申鐃暑。(鐃緒申鐃書式緒申鐃緒申砲弔鐃緒申鐃� $T>U, T>V$ 鐃叔わ申鐃緒申蠅�鐃淳ワ申鐃叔わ申鐃緒申)鐃淑わ申鐃緒申鐃緒申任茲わ申鐃緒申鐃緒申鐃緒申鐃薯見てわ申蕕�鐃緒申鐃夙にわ申鐃銃￥申 鐃夙りあ鐃緒申鐃緒申鐃熟計誌申鐃緒申鐃銃みわ申伐鐃緒申鐃緒申辰討鐃緒申襪�分鐃緒申鐃暑か鐃緒申鐃塾わ申覆鐃緒申鐃�)

3.
$\operatorname{Image}({}^a\phi)$ 鐃塾鰹申鐃緒申鐃緒申鐃緒申覆鐃緒申鐃緒申鐃� ( $t\to \pm \infty$, $t\to 1$ , $t\to \pm 1$ 鐃塾誌申鐃祝どわ申鐃淑るか鐃祝とわ申鐃緒申鐃緒申鐃春わ申 払鐃緒申鐃緒申鐃緒申)

(鐃緒申鐃緒申)

(1)

\begin{displaymath}{}^a\phi(t)=(t^2,t(t^2-1)) \end{displaymath}

鐃夙わ申鐃緒申鐃緒け鐃叔わ申鐃暑か鐃初、 $t=0,1,2,-1$ 鐃薯それぞ鐃緒申鐃緒申鐃緒申鐃緒申鐃銃￥申

\begin{displaymath}{}^a\phi(0)=(0,0),\quad {}^a\phi(1)=(1,0),\quad {}^a\phi(2)=(4,6),\quad {}^a\phi(-1)=(1,0) \end{displaymath}

鐃夙わ申鐃緒申鐃緒申鐃祝なる。

(2) 鐃述ワ申箸砲鐃緒申鐃処う鐃祝￥申

\begin{displaymath}U=T^2, V=T(T^2-1) \quad(\phi(X)=U ,\phi(Y)=V \text { と書いた}) \end{displaymath}

鐃緒申鐃緒申 $T$ 鐃緒申探遒刻申鐃出よい鐃緒け鐃緒申鐃緒申

\begin{displaymath}V^2=U(U-1)^2 \end{displaymath}

鐃醇え鐃祝￥申

\begin{displaymath}\phi(Y^2-X(X-1)^2)=0 \end{displaymath}

鐃緒申辰董鐃� $Y^2-X(X-1)^2$ 鐃緒申 $\ker(\phi)$ 鐃緒申鐃緒申鐃緒申里鐃緒申里襦� 鐃緒申鐃夙わ申

\begin{displaymath}\ker(\phi)=(Y^2-X(X-1)^2) \end{displaymath}

鐃緒申里鐃緒申鐃緒申个茲わ申錣縁申鐃緒申鐃緒申鐃緒申鐃緒申鐃祝わ申鐃純え鐃緒申 $Y$ 鐃緒申鐃緒申鐃旬わ申鐃緒申鐃緒申蟷誌申鐃� 鐃峻わ申鐃緒申法鐃緒申箸鐃緒申个茲わ申鐃緒申鐃緒申鐃緒申鐃緒申楮戮蝋峙鐃緒申撚鐃緒申發件申鐃緒申里任鐃緒申鐃緒申任鐃塾�鐃緒申鐃暑。

(3) (鐃緒申略)

鐃緒申鐃緒申 1.4  

$A=\mbox{${\Bbb R}$ }[X,Y]$ 鐃塾ワ申鐃叔ワ申鐃緒申 $I=(X^2-4Y^2-1)$ 鐃緒申 $B=\mbox{${\Bbb R}$ }[T]_{(T^2-1)} $ 鐃夙につわ申鐃銃￥申 $\mbox{${\Bbb R}$ }$-鐃緒申同鐃緒申 $\phi:A\to B$ 鐃緒申

\begin{displaymath}\phi(X)=\frac{1+T^2}{1-T^2},\quad \phi(Y)=\frac{T}{1-T^2} \end{displaymath}

鐃緒申鐃緒申鐃緒申鐃緒申襦ｏ申鐃緒申里箸鐃緒申鐃�
1.
$V(I)$ 鐃緒申渕鐃緒申鐃緒申覆鐃緒申鐃緒申鐃�
2.
$\phi(I)=0$ 鐃叔わ申鐃緒申鐃緒申鮗┐鐃緒申覆鐃緒申鐃緒申鐃�
3.
鐃緒申鐃緒申鐃縦の誌申鐃緒申鐃初、 $\phi$ 鐃緒申 $\mbox{${\Bbb R}$ }\setminus\{\pm 1\}$鐃緒申鐃緒申 $V(I)$ 鐃舜の種申鐃緒申 $f$ 鐃緒申鐃緒申鐃暑こ鐃夙わ申分鐃緒申鐃暑が鐃緒申 鐃緒申鐃緒申 $f$ 鐃祝つわ申鐃銃￥申

\begin{displaymath}f(0), f(\pm 2),f( \pm 3) \end{displaymath}

鐃薯それぞ鐃緒申鐃潤、鐃緒申鐃緒申砲鐃緒申鐃緒申(1) 鐃塾ワ申鐃緒申佞暴颪�刻申鐃淳なわ申鐃緒申鐃緒申

(鐃緒申鐃緒申)

鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃�11.2 鐃夙ほわ申同鐃緒申鐃叔わ申鐃暑。(鐃緒申辰半椶鐃緒申鐃緒申鐃緒申鐃緒申弌鐃�11.2 鐃緒申単鐃緒申 $Y$ 鐃緒申鐃緒申鐃緒申 $1/2$ 鐃緒申鐃駿わ申鐃緒申鐃緒申鐃緒申鐃叔わ申鐃暑。)鐃緒申鐃緒申鐃緒申鐃獣て駕申鐃緒申鐃熟常申略鐃緒申鐃暑。

next up previous
Next: About this document ...
Yoshifumi Tsuchimoto
2001-07-31