Motivic zeta function

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In algebraic geometry, the motivic zeta function of a smooth algebraic variety ${\displaystyle X}$ is the formal power series:^[1]

${\displaystyle Z(X,t)=\sum _{n=0}^{\infty }[X^{(n)}]t^{n}}$

Here ${\displaystyle X^{(n)}}$ is the ${\displaystyle n}$-th symmetric power of ${\displaystyle X}$, i.e., the quotient of ${\displaystyle X^{n}}$ by the action of the symmetric group ${\displaystyle S_{n}}$, and ${\displaystyle [X^{(n)}]}$ is the class of ${\displaystyle X^{(n)}}$ in the ring of motives (see below).

If the ground field is finite, and one applies the counting measure to ${\displaystyle Z(X,t)}$, one obtains the local zeta function of ${\displaystyle X}$.

If the ground field is the complex numbers, and one applies Euler characteristic with compact supports to ${\displaystyle Z(X,t)}$, one obtains ${\displaystyle 1/(1-t)^{\chi (X)}}$.