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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Motivic zeta function ： ウィキペディア英語版 Motivic zeta function In algebraic geometry, the motivic zeta function of a smooth algebraic variety $X$ is the formal power series :$Z(X,t)=\backslash sum\_^\backslash infty\; ()t^n$ Here $X^$ is the $n$-th symmetric power of $X$, i.e., the quotient of $X^n$ by the action of the symmetric group $S\_n$, and $()$ is the class of $X^$ in the ring of motives (see below). If the ground field is finite, and one applies the counting measure to $Z(X,t)$, one obtains the local zeta function of $X$. If the ground field is the complex numbers, and one applies Euler characteristic with compact supports to $Z(X,t)$, one obtains $1/(1-t)^$. == Motivic measures == A motivic measure is a map $\backslash mu$ from the set of finite type schemes over a field $k$ to a commutative ring $A$, satisfying the three properties :$\backslash mu(X)\backslash ,$ depends only on the isomorphism class of $X$, :$\backslash mu(X)=\backslash mu(Z)+\backslash mu(X\backslash setminus\; Z)$ if $Z$ is a closed subscheme of $X$, :$\backslash mu(X\_1\backslash times\; X\_2)=\backslash mu(X\_1)\backslash mu(X\_2)$. For example if $k$ is a finite field and $A=$ is the ring of integers, then $\backslash mu(X)=\backslash \#(X(k))$ defines a motivic measure, the ''counting measure''. If the ground field is the complex numbers, then Euler characteristic with compact supports defines a motivic measure with values in the integers. The zeta function with respect to a motivic measure $\backslash mu$ is the formal power series in $At$ given by :$Z\_\backslash mu(X,t)=\backslash sum\_^\backslash infty\backslash mu(X^)t^n$. There is a ''universal motivic measure''. It takes values in the K-ring of varieties, $A=K(V)$, which is the ring generated by the symbols $()$, for all varieties $X$, subject to the relations :$()=()\backslash ,$ if $X\text{'}$ and $X$ are isomorphic, :$()=()+(Z)$ if $Z$ is a closed subvariety of $X$, :$(X\_2)=()\backslash cdot()$. The universal motivic measure gives rise to the motivic zeta function. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Motivic zeta function」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.