Open Problems in Arithmetic Dynamics — Conjecture 51

Definitions and mathematical statement

Let $f\in{\mathbb{Q}}(x)$ be a rational function of degree at least $2$, regarded as a map $f:{\mathbb{P}}^1({\mathbb{Q}})\to{\mathbb{P}}^1({\mathbb{Q}})$. If $f=P/Q$ with coprime polynomials $P,Q\in{\mathbb{Q}}[x]$, its degree is $\max\{\deg P,\deg Q\}$. Write $f^0$ for the identity and $f^{n+1}=f\circ f^n$. A point $\gamma\in{\mathbb{P}}^1({\mathbb{Q}})$ is $f$-wandering if its forward orbit $\{f^n(\gamma):n\geq 0\}$ is infinite.

Conjecture 51, as supplied. For every rational function $f\in{\mathbb{Q}}(x)$ of degree at least $2$ and every pair of $f$-wandering points $\alpha,\beta\in{\mathbb{Q}}$, the Zsigmondy set of $\mathcal A(f,\alpha,\beta)$ is finite: \[ \#\operatorname{Zsig}\left(\mathcal A(f,\alpha,\beta)\right)<\infty. \] Unresolved definition. The supplied record does not define $\mathcal A(f,\alpha,\beta)$, its indexing, or the primitive-divisor convention used to form its Zsigmondy set. For a sequence $(a_n)_{n\geq 1}$ of nonzero integers, the usual convention calls a prime $p$ a primitive prime divisor of $a_n$ if $p\mid a_n$ and $p\nmid a_m$ for every $1\leq m<n$; the corresponding Zsigmondy set consists of indices whose terms have no primitive prime divisor. This explains the terminology only and does not identify the unspecified object $\mathcal A(f,\alpha,\beta)$ with such a sequence. Its construction and conventions must be recovered from the cited source before the conjecture has a fully self-contained formulation.

Short English statement

The supplied conjecture says that every rational map of degree at least two and every pair of rational wandering points give an object $\mathcal A(f,\alpha,\beta)$ with finite Zsigmondy set. The record omits the definition of this object and its primitive-divisor conventions, so the precise arithmetic assertion remains incomplete.

Sources

• Local UnsolvedMath registry: permanent ID $800428$, source problem code catalogue entry, title Open Problems in Arithmetic Dynamics --- Conjecture 51. The supplied registry labels the problem open; this is a supplied status, not an independently verified literature assessment.
• Benedetto, DeMarco, Ingram, Jones et al., Current Trends and Open Problems in Arithmetic Dynamics (2018), Conjecture 51. Attribution and proposed year are supplied by the registry. Source URL: https://arxiv.org/abs/1806.04980.
• Supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance statements; the source has not been independently accessed or verified for this draft.