Open Problems in Arithmetic Dynamics — Question 41

Definitions and mathematical statement

Fix an integer $N\geq 1$ and a prime power $q$, and put \[ X={\mathbb{P}}^N({\mathbb{F}}_q),\qquad m=\#X=1+q+\cdots+q^N. \] The supplied excerpt does not define its graph and averaging notation. For this draft, use the following conventional interpretations, which require confirmation against the source context. For a positive integer $d$, let $\operatorname{End}_d^N({\mathbb{F}}_q)$ denote the set of degree-$d$ morphisms $f:{\mathbb{P}}^N\to{\mathbb{P}}^N$ defined over ${\mathbb{F}}_q$. Such a morphism is represented by $N+1$ homogeneous degree-$d$ polynomials over ${\mathbb{F}}_q$ having no common zero in ${\mathbb{P}}^N(\overline{{\mathbb{F}}}_q)$, with tuples identified under multiplication by a common nonzero scalar. Distinct morphisms are counted separately even when they induce the same function on $X$.

For a function $g:X\to X$, let $\Gamma_g$ be the directed graph with vertex set $X$ and one edge $x\to g(x)$ for each $x\in X$. Let $\operatorname{Comp}(\Gamma_g)$ count connected components after forgetting edge directions. For an integer $m\geq1$, define the uniform random-function average by \[ \operatorname{Comp}_{\mathrm{avg}}(m) =\frac{1}{m^m}\sum_{g:\{1,\ldots,m\}\to\{1,\ldots,m\}} \operatorname{Comp}(\Gamma_g). \] For an endomorphism $f$, its graph means the graph of its restriction to $X$.

Under these interpretations, the supplied Question 41 asks whether \[ \frac{1}{\#\operatorname{End}_d^N({\mathbb{F}}_q)} \sum_{f\in\operatorname{End}_d^N({\mathbb{F}}_q)} \operatorname{Comp}(\Gamma_f) =\operatorname{Comp}_{\mathrm{avg}}(m)+O(1) \qquad(d\to\infty). \] With $N$ and $q$ fixed as stated, the literal bounded-error formulation is that there exist $C_{N,q}\geq0$ and $d_0(N,q)\geq1$ such that, for every integer $d\geq d_0(N,q)$, \[ \left| \frac{1}{\#\operatorname{End}_d^N({\mathbb{F}}_q)} \sum_{f\in\operatorname{End}_d^N({\mathbb{F}}_q)} \operatorname{Comp}(\Gamma_f) -\operatorname{Comp}_{\mathrm{avg}}(m) \right|\leq C_{N,q}. \] The excerpt does not specify whether the error bound must instead be uniform in $N$ or $q$. Because the vertex set is fixed in the stated limit, the intended strength of this comparison requires source-context review; no uniformity or alternative limiting regime is imposed here.

Short English statement

For fixed projective dimension $N$ and finite field ${\mathbb{F}}_q$, compare the average number of graph components of degree-$d$ projective endomorphisms with the average for uniformly chosen functions on the same number of points, asking for bounded error as $d\to\infty$. The notation and intended uniformity of the bound require confirmation. The registry labels this question open; that label is supplied metadata, not an independently verified assessment.

Sources

• Local UnsolvedMath registry, permanent ID $800418$, exact source problem code catalogue entry, title "Open Problems in Arithmetic Dynamics --- Question 41." The supplied record attributes the problem to Benedetto, DeMarco, Ingram, Jones et al., with proposed year 2018 and status open.
• Supplied source: Benedetto, DeMarco, Ingram, Jones et al., Current Trends and Open Problems in Arithmetic Dynamics (2018), Question 41, https://arxiv.org/abs/1806.04980. The provenance records access on 2026-10-16 and extraction method arxiv_latex. These are supplied provenance details; the source and current mathematical status have not been independently verified for this draft.