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Let $f(x,y), g(x,y)$ be two polynomials with integral coefficients, $K$ be the algebraic closure of $\mathbb{Z}/p\mathbb{Z}$. A point $(a,b)\in K\times K$ is called quasi-fixed of degree $k$ if $f(a,b)=a^{p^k}, g(a,b)=b^{p^k}$ (in other words, the orbit of the Frobenius map containing $(a,b)$ is stable under $(f,g)$). A. Borisov and I proved that the set of quasi-fixed points is Zariski dense for almost all $p$, see: Borisov, Alexander; Sapir, Mark, Polynomial maps over finite fields and residual finiteness of mapping tori of group endomorphisms. Invent. Math. 160 (2005), no. 2, 341–356.

Let $N_k$ be the number of quasi-fixed points of degree $k$.

Is the function $\sum_k N_k n^k$ rational?

A more general question (when the number of variables is $\ge 3$) was asked by Grunewald. As far as I know it is open, but in the case of two variables, it might be easier. If $(f,g)$ is a finite map (the pre-image of every point is finite), the result follows from the Lefschetz trace formula proved by Pink and Fujiwara.

Update: In fact the result should be true for every algebraic scheme $M$ over $F_p$ and every self-map $f: M\to M$ (one should even be able to consider a correspondence instead of a map). If the map is the identity, the fact follows from the Weil conjecture (proved by Deligne). If the map (correspondence) is finite, it follows from Deligne conjecture (proved by Pink and Fujiwara). The fact that the set of quasi-fixed points is Zariski dense provided $F$ is dominant was proved by A. Borisov and myself in the case when $M=A^n$, and by Hrushovsky for general $M$. I specified the case $M=A^2$ (the scheme is affine of dim 2) because that case was much simpler than the general case.

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Could you define "dominant" in this context? – Bill Thurston Dec 30 '10 at 0:10
Dominant map = image is Zariski dense. – Mark Sapir Dec 30 '10 at 0:20

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