Open Problems in Arithmetic Dynamics — Question 42

Definitions and mathematical statement

Let $f:{\mathbb{P}}^N\to{\mathbb{P}}^N$ be a morphism defined over ${\mathbb{Q}}$, of degree $d\ge 2$. The supplied record does not explicitly specify the range of the dimension $N$. For a prime $p$ of good reduction, write $\widetilde f_p$ for the reduced morphism and set $X_p={\mathbb{P}}^N({\mathbb{F}}_p)$. The functional graph $\Gamma_{\widetilde f_p}$ has vertex set $X_p$ and a directed edge $x\to\widetilde f_p(x)$ for every $x\in X_p$.

Unresolved notation. The supplied record does not define the graph statistic $\operatorname{Comp}$, the benchmark $\operatorname{Comp}_{\mathrm{avg}}$, or the averaging model underlying that benchmark. These symbols are retained without assigning them an interpretation. In particular, the benchmark is evaluated at $p^{(1-\epsilon)N}$, which need not be an integer and, for $\epsilon>1$ and $N>0$, is less than one. Its domain or any rounding convention is not supplied.

Subject to resolving these definitions, the source asks whether, for every such $f$ and every real $\epsilon>0$, the set \[ S_{f,\epsilon}=\left\{p:\operatorname{Comp}(\Gamma_{\widetilde f_p})\leq \operatorname{Comp}_{\mathrm{avg}}\!\left(p^{(1-\epsilon)N}\right)\right\} \] has density zero among the primes. The source does not specify the type of density. If natural density among primes is intended, this means \[ \lim_{B\to\infty}\frac{\#\{p\leq B:p\in S_{f,\epsilon}\}}{\#\{p\leq B:p\text{ is prime}\}}=0. \] Here restricting the set to primes of good reduction is an explicit conventional interpretation of the reduction notation; the supplied statement does not state its prime exclusions or choose an integral model.

The source also asks the analogous question with ${\mathbb{A}}^N$ in place of ${\mathbb{P}}^N$. Under the interpretation that $f:{\mathbb{A}}^N\to{\mathbb{A}}^N$ is a morphism defined over ${\mathbb{Q}}$, the corresponding graph has vertex set ${\mathbb{A}}^N({\mathbb{F}}_p)$ and edges $x\to\widetilde f_p(x)$, wherever reduction is defined. The intended affine class of maps and degree convention require clarification; the phrase "similarly" does not specify them.

Finally, the source asks for which maps $f$ stronger conclusions can be obtained, giving as an example \[ \operatorname{Comp}(\Gamma_{\widetilde f_p})\asymp\operatorname{Comp}_{\mathrm{avg}}(p^N) \qquad\text{as }p\to\infty. \] With the usual meaning of $\asymp$, this requests positive constants $c_f,C_f$ such that, for every sufficiently large admissible prime $p$, \[ c_f\operatorname{Comp}_{\mathrm{avg}}(p^N)\leq\operatorname{Comp}(\Gamma_{\widetilde f_p})\leq C_f\operatorname{Comp}_{\mathrm{avg}}(p^N). \] This is a question about identifying maps with stronger behavior, not an assertion that the displayed comparison holds for all maps. The missing definitions and conventions above prevent a fully specified mathematical formulation.

Short English statement

For a rationally defined projective morphism of degree at least two, the source asks whether primes at which a graph statistic falls below a specified average benchmark have density zero, for every $\epsilon>0$. It asks the analogous affine question and seeks maps satisfying stronger comparisons. The statistic, its average benchmark, the density convention, and parts of the affine formulation are unspecified in the supplied record.

Sources

• Local UnsolvedMath registry, permanent ID $800419$, exact source problem code catalogue entry, title "Open Problems in Arithmetic Dynamics --- Question 42." The supplied registry labels the problem open; this label is source metadata, not an independently verified mathematical or literature assessment.
• Benedetto, DeMarco, Ingram, Jones et al., Current Trends and Open Problems in Arithmetic Dynamics (2018), Question 42. Supplied source URL: https://arxiv.org/abs/1806.04980. Attribution and proposed year are supplied by the registry.
• Supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. This access date is supplied provenance, not a claim of independent source consultation or verification. The source context defining the graph statistic and its average was not included in the supplied record.