# Bounding the number of "generalized $\mathbb{F}_q$-rational points" of a variety in terms of dimension and degree

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.