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鐃緒申鐃緒申鐃� II 鐃緒申鐃緒申 No.1

\fbox{本講義の目的}

鐃緒申鐃塾常申鐃渋随申犲逸申弔離鐃緒申妊鐃緒申鐃竣とわ申鐃緒申鐃緒申鐃緒申鐃緒申鐃重�鐃祝醐申鐃緒申鐃緒申鐃塾わ申鐃緒申 鐃渋わ申鐃緒申鐃重�鐃緒申鐃緒申噺討个鐃暑集鐃緒申鐃緒申弍鐃緒申鐃緒申討鐃緒申襪鰹申箸鐃緒申鐃緒申鬚掘鐃� 鐃緒申鐃緒申鐃出縁申鐃舜件申鐃祝わ申辰謄鐃緒申妊鐃緒申鐃塾種々鐃緒申鐃緒申鐃緒申鐃塾わ申聴鐃縮ｏ申鐃縦常申鐃重�鐃緒申 鐃夙らえ鐃緒申鐃緒申茲�鐃祝なる。

鐃緒申鐃緒申鐃塾テ￥申鐃緒申:

\fbox{代数的集合(多項式の共通零点)}

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�C(鐃緒申鐃緒申),鐃緒申鐃緒申鐃�I 鐃叔器申鐃緒申鐃塾ってわ申鐃緒申呂鐃緒申覆里如鐃� 忘鐃曙た鐃粛わ申鐃緒申鐃緒申鐃緒申鐃銃わ申鐃緒申鐃銃駕申鐃緒申鐃緒申鐃緒申

鐃緒申鐃� 1.1 (鐃緒申鐃緒申)  
1.
鐃縦とは￥申鐃緒申鐃塾なわ申鐃叔加醐申鐃処算鐃緒申鐃緒申由鐃祝でわ申鐃緒申茲�鐃淑緒申鐃緒申任鐃緒申襦�
2.
鐃塾とは￥申鐃縦でわ申鐃獣てわ申鐃緒申鐃� $0$ 鐃淑鰹申鐃塾醐申鐃塾逆醐申鐃緒申鐃殉わ申 鐃緒申鐃緒申鐃緒申亡泙泙鐃緒申茲�鐃淑緒申鐃緒申任鐃緒申襦�

鐃緒申 1.1  
1.
有鐃緒申鐃緒申鐃緒申鐃塾の緒申鐃緒申 $\mbox{${\Bbb Q}$ }$, 鐃渋随申鐃緒申鐃塾の緒申鐃緒申 $\mbox{${\Bbb R}$ }$, 複鐃叔随申鐃緒申鐃塾の緒申鐃緒申 ${\Bbb C}$ 鐃熟わ申鐃曙ぞ鐃緒申鐃塾でわ申鐃暑。
2.
鐃緒申鐃緒申鐃緒申鐃塾の緒申鐃緒申 ${\mbox{${\Bbb Z}$ }}$ 鐃熟環ではわ申鐃暑が鐃緒申鐃塾ではなわ申鐃緒申
3.
鐃緒申鐃緒申鐃緒申鐃緒申鐃塾の緒申鐃緒申 $\Bbb N$ 鐃熟環でわ申鐃緒申覆鐃緒申鐃�

鐃緒申鐃� 1.2   鐃緒申 $k$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 $n$ 鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申襦ｏ申鐃緒申里箸鐃緒申鐃� $k$ 鐃塾醐申鐃薯係随申鐃祝わ申鐃� $X_1,...X_n$ 鐃緒申多鐃準式鐃緒申鐃緒申鐃塾のなわ申鐃緒申鐃緒申鐃� 鐃縦になる。 鐃緒申鐃塾環のわ申鐃夙わ申$k$ 鐃緒申鐃� $n$ 鐃術随申多鐃準式鐃縦とわ申咫鐃� $k[X_1,X_2,\dots,X_n]$ 鐃緒申 鐃緒申鐃緒申錣刻申鐃� (鐃緒申鐃塾講居申鐃叔はわ申鐃緒申鐃� $k$ 鐃緒申鐃塾の常申鐃薯扱わ申鐃緒申鐃緒申鐃緒申鐃祝わ申辰討鐃� $k$ 鐃緒申鐃縦の常申鐃緒申 余鐃熟わ申弔鐃緒申討鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申茲わ申鐃�)

(鐃緒申鐃緒申)

鐃術随申鐃塾なまわ申鐃緒申必鐃緒申鐃緒申鐃緒申 $X_1,X_2,\dots,X_n$ 鐃祝わ申鐃淑わ申鐃緒申个覆鐃淑わ申鐃緒け鐃叔はなわ申鐃緒申 鐃緒申鐃夙わ申鐃緒申鐃緒申鐃術随申($n=2$) 鐃塾誌申鐃祝わ申 $X,Y$, 3鐃術随申鐃塾誌申鐃祝わ申 $X,Y,Z$ 鐃緒申鐃術随申鐃緒申鐃術わ申鐃緒申鐃緒申鐃緒申 添鐃緒申鐃緒申颪�なわ申鐃銃よい分鐃緒申鐃緒申鐃准わ申鐃緒申鐃緒申鐃緒申

鐃緒申鐃� 1.3   鐃緒申 $k$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 $n$ 鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申襦� $k[X_1,X_2,\dots,X_n]$ 鐃緒申鐃緒申分鐃緒申鐃緒申 $T$ 鐃緒申鐃出わ申鐃銃￥申 $k^n$ 鐃緒申鐃緒申分鐃緒申鐃緒申

\begin{displaymath}V(T)= \left \{ x=(x_1,x_2,\dots,x_n)\in k^n \left \vert \quad f(x)=0 \qquad (\forall f\in T) \right . \right \} \end{displaymath}

鐃緒申 $T$ 鐃塾駈申鐃緒申鐃緒申鐃緒申鐃塾緒申鐃緒申噺討屐鐃�

鐃緒申 1.2   $k=\mbox{${\Bbb R}$ }$ 鐃叔考わ申鐃銃￥申$n=2$ 鐃塾常申鐃祝￥申
1.
$V(\{X\})$ 鐃緒申 Y鐃緒申鐃叔わ申鐃暑。
2.
$V(\{X,Y\})$ 鐃緒申 $V(\{X-5,Y-2\})$ 鐃熟逸申鐃緒申鐃緒申鐃緒申覆襦�
3.
$V(\{Y-X^2\})$ 鐃緒申鐃緒申物鐃緒申鐃叔わ申鐃暑。
4.
$V(\{X^2-Y^2\})$ 鐃緒申鐃緒申直鐃緒申鐃緒申鐃出ッテワ申妨鐃緒申辰鐃緒申鐃塾でわ申鐃暑。
5.
$V(\{X^2-Y^2,Y-1\})$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申覆觸醐申鐃� $\{(1,1),(-1,1)\}$ 鐃叔わ申鐃暑。
6.
$V(\{X^2+Y^2-1\})$ 鐃熟円￥申 $V(\{X^2+Y^2\})$ 鐃熟醐申鐃緒申鐃緒申 $V(\{X^2+Y^2+1\})$ 鐃熟駈申鐃緒申鐃緒申任鐃緒申襦�

鐃緒申 1.3   $k=\mbox{${\Bbb R}$ }$ 鐃叔考わ申鐃銃￥申$n=3$ 鐃塾とわ申鐃祝￥申
1.
$V(\{X^2+Y^2+Z^2-1\})$ 鐃熟醐申鐃緒申鐃緒申鐃醇心鐃夙わ申鐃緒申鐃緒申鐃縮でわ申鐃暑。
2.
$V(\{X^2+Y^2-Z^2\})$ 鐃熟円随申鐃叔わ申鐃暑。

鐃緒申鐃緒申 1.1   $k=\mbox{${\Bbb R}$ }$ 鐃緒申 $n=2$ 鐃塾とわ申鐃祝つわ申鐃銃￥申 $\mbox{${\Bbb R}$ }[X,Y]$ 鐃緒申鐃緒申分鐃緒申鐃緒申 $S$ 鐃緒申鐃緒申鐃� 鐃緒申分鐃叔考わ申鐃銃￥申鐃緒申鐃緒申 $S$ 鐃緒申鐃出わ申鐃緒申 $V(S)$ 鐃緒申渕鐃緒申鐃緒申覆鐃緒申鐃緒申鐃� (鐃淑わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申佞砲覆鐃塾わ申鐃緒申鐃緒申単鐃緒申鐃緒申由鐃盾そ鐃緒申鐃暑こ鐃緒申) (鐃緒申鐃殉わ申鐃書しわ申鐃緒申鐃粛わ申鐃緒申鐃銃わ申鐃緒申鐃緒申鐃緒申鐃緒申僂任鐃緒申襪�鐃緒申 鐃所頃鐃塾わ申里任茲わ申鐃緒申鐃緒申鐃緒申覆鐃緒申箸鐃渋常申凌佑硫鐃緒申鐃緒申判鼎覆鐃淑わ申鐃緒申鐃初い 鐃塾関随申鐃緒申鐃術わ申鐃緒申茲�鐃緒申鐃緒申鐃熟わ申鐃暑こ鐃夙￥申)


Yoshifumi Tsuchimoto
2001-04-16