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In what follows, for a prime power $q=p^m$, $\phi_q$ denotes the Frobenius endomorphism $x\mapsto x^q$ of a finite-dimensional affine space over the algebraic closure $\overline{\mathbb{F}_p}$ (the dimension of the space will be clear from the context). My question is the following:

Question:

Given positive integers $n,d,e,f$, does there exist a (preferrably effective) constant $B=B(n,d,e,f)$ with the following property:

For every prime $p$, every power $q=p^m$, every positive integer $t$ and every affine variety $V\subseteq\overline{\mathbb{F}_p}^n$ of dimension at most $d$ and degree at most $t$ defined over $\overline{\mathbb{F}_p}$, and for every endomorphism $h:V\rightarrow V$ defined (in $V^2$) over $\overline{\mathbb{F}_p}$ by at most $e$ polynomial equations of degree at most $f$, we have

$|\{a\in V\mid \phi_q(a)=h(a)\}|\leq B\cdot t\cdot q^d$.

For those interested, let me provide some motivation for this algebro-geometric question from a group-theoretic point of view, also explaining the title.

In Section 4 of their paper [1], Dixon, Pyber, Seress and Shalev use the following algebro-geometric lemma of Hrushovski to give upper bounds on the number of solutions to an equation of the form $w=1$, $w$ a reduced word, in finite simple groups of Lie type:

Lemma (Hrushovski):

Given positive integers $d,e,f,n$, there exists a constant $B=B(d,e,f,n)$ with the following property:

For every prime $p$, every power $q=p^m$, and for every affine variety $V\subseteq\overline{\mathbb{F}_p}^n$ of dimension at most $d$, defined over $\overline{\mathbb{F}_p}$ by at most $e$ polynomial equations of degree at most $f$, and for every endomorphism $h:V\rightarrow V$ defined (in $V^2$) over $\overline{\mathbb{F}_p}$ by at most $e$ polynomial equations of degree at most $f$, we have

$|\{a\in V\mid \phi_q(a)=h(a)\}|\leq B\cdot q^d$.

In their application, $V$ is the subvariety, cut out by the word equation $w=1$ in $r$ distinct variables, of $G^r$, where $G=G(\overline{\mathbb{F}_p})$ is a simple Chevalley group, and the endomorphism $h$ is chosen such that $G_q=\{a\in G\mid\phi_q(a)=h(a)\}$ is a finite group containing one of the finite simple groups of Lie type as a subgroup of bounded index.

As the four authors say themselves, "usually", $h$ can be chosen as the identity function, so that passing to the subvariety cut out by $\phi_q(a)=h(a)$ just corresponds to restricting to the $\mathbb{F}_q$-rational points, but the generalization is necessary to include the Ree and Suzuki families. Without this complication, one could simply use the following Lemma, Claim 7.2 in [2], to get an upper bound with the same asymptotic behavior:

Lemma (Dvir-Kollár-Lovett):

For any finite field $\mathbb{F}$ and any variety $V$ in $\overline{\mathbb{F}}^n$ of degree $t$ and dimension $d$, we have $|V\cap \mathbb{F}^n|\leq t\cdot|\mathbb{F}|^d$.

The nice feature of this result is that the constant in front of $|\mathbb{F}|^d$ is explicit and grows linearly in the degree of $V$. My question asks whether the same can be achieved in the more general situation from Hrushovski's Lemma (assuming that the ambient dimension $n$ as well as the number and the degrees of the equations defining $h$ are bounded).

References:

[1] J.D. Dixon, L. Pyber, Á. Seress, A. Shalev, Residual properties of free groups and probabilistic methods. J. Reine Angew. Math. (Crelle's) 556 (2003), 159-172.

[2] Z. Dvir, J. Kollár, S. Lovett, Variety Evasive Sets. Comput. Complex. 23(4) (2014), 509-529.

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