鐃緒申鐃緒申鐃�II 鐃緒申鐃緒申 No.10

鐃淑駕申

鐃緒申鐃叔随申鐃叔わ申鐃緒申箸鐃緒申鐃�

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鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申)鐃叔わ申鐃緒申箸鐃緒申襦�

鐃緒申鐃緒申鐃緒申鐃緒申

鐃縦わ申鐃術随申 $X_1,\dots,X_n$ 鐃祝関わ申鐃緒申 $% latex2html id marker 797 $ {\mathbb{F}}_q$$ 鐃緒申鐃緒申鐃緒申多鐃準式 $f_1,f_2,\dots,f_m$ 鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申鐃� 鐃緒申鐃緒申鐃緒申鐃緒申

	$\displaystyle f_1(X_1,X_2,\dots,X_n)=0,$
	$\displaystyle f_2(X_1,X_2,\dots,X_n)=0,$
	$\displaystyle \dots,$
	$\displaystyle \dots,$
	$\displaystyle \dots,$
	$\displaystyle f_m(X_1,X_2,\dots,X_n)=0$

$% latex2html id marker 813 $ {\mathbb{F}}_{q^r}$$ 鐃叔わ申 $V(f_1,\dots,f_m)$ 鐃塾駕申鐃緒申鐃緒申里鐃� $% latex2html id marker 817 $ V(f_1,\dots,f_m)({\mathbb{F}}_{q^r})$$ 鐃叔緒申表鐃緒申鐃緒申

鐃緒申鐃� 10.1 鐃緒申鐃緒申鐃緒申鐃緒申

鐃塾刻申同鐃緒申鐃緒申鐃緒申鐃舜随申鐃緒申

$% latex2html id marker 822 $\displaystyle Z(V,t)= \exp \left ( \sum_{k=1}^\infty \frac{\char93 V({\mathbb{F}}_{q^k})}{k}t^k \right) $$

鐃祝わ申辰鐃緒申鐃緒申鐃緒申鐃暑。鐃緒申鐃緒申鐃� $% latex2html id marker 824 $ {\mathbb{F}}_q$$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃熟￥申 $% latex2html id marker 826 $ Z(V/{\mathbb{F}}_q,t)$$ 鐃淑どとわ申颪�鐃�

鐃緒申鐃緒申鐃緒申鐃熟器申分鐃緒か鐃緒申砲鐃緒申鐃緒申鐃緒申鐃緒申里鐃淑わ申鐃緒申鐃緒申鐃塾種申鐃緒申 $\exp$ 鐃緒申鐃渋は件申未鐃粛㍼申鐃緒申鐃緒申砲鐃緒申襪随申鐃塾刻申鐃竣でわ申鐃暑。鐃緒申鐃殉￥申 $% latex2html id marker 830 $ \char93 V({\mathbb{F}}_{q^k})=\operatorname{tr}(A_1^k)-\operatorname{tr}(A_2)^k$$ 鐃夙なわ申茲�申聞鐃緒申鐃�

鐃緒申存鐃淳わ申鐃緒申鐃夙わ申鐃緒申覆鐃出￥申

	$% latex2html id marker 833 $\displaystyle \exp \left ( \sum_{k=1}^\infty \frac{\char93 V({\mathbb{F}}_{q^k})}{k}t^k \right)$$
$\displaystyle =$	$\displaystyle \exp \left ( \sum_{k=1}^\infty \frac{\operatorname{tr}(A_1^k)-\operatorname{tr}(A_2^k)}{k}t^k \right)$
$\displaystyle =$	$\displaystyle \exp( \operatorname{tr}(-\log(1-A_1t)-\operatorname{tr}(-\log(1-A_2t))$
$\displaystyle =$	$\displaystyle \operatorname{det}(1-A_2t)/\operatorname{det}(1-A_1t)$