Preperiods and Cycles of Quadratic Polynomial Iteration

Definitions and mathematical statement

Let $q=p^r$, where $p$ is prime and $r\geq 1$, and let ${\mathbb{F}}_q$ be the field with $q$ elements. Let \[ f(x)=ax^2+bx+c\in{\mathbb{F}}_q[x],\qquad a\neq 0, \] and let $m\in{\mathbb{F}}_q$. Define $x_0=m$ and $x_{n+1}=f(x_n)$ for every integer $n\geq 0$; thus $x_n=f^{\circ n}(m)$, with $f^{\circ 0}$ the identity map. Since ${\mathbb{F}}_q$ is finite, this sequence is eventually periodic. Define its preperiod by \[ \mu(f,m)=\min\left\{u\in{\mathbb{Z}}_{\geq 0}:\text{ there exists }v\in{\mathbb{Z}}_{\geq 1}\text{ such that }x_{u+v}=x_u\right\}, \] and its eventual period by \[ \lambda(f,m)=\min\left\{v\in{\mathbb{Z}}_{\geq 1}:x_{\mu(f,m)+v}=x_{\mu(f,m)}\right\}. \] The eventual cycle consists of the $\lambda(f,m)$ distinct elements beginning at $x_{\mu(f,m)}$, and $x_{n+\lambda(f,m)}=x_n$ for every $n\geq\mu(f,m)$. In particular, $\mu(f,m)=0$ when the initial value already lies on a cycle.

Problem. For every prime power $q$, every quadratic polynomial $f\in{\mathbb{F}}_q[x]$, and every $m\in{\mathbb{F}}_q$, determine the exact values of $\mu(f,m)$ and $\lambda(f,m)$ in terms of the given data.

Unresolved qualification. The supplied statement imposes no restriction on the characteristic or on the quadratic family. It does not specify whether the requested determination should be an explicit formula, a structural classification, or an algorithm with prescribed efficiency; in particular, it does not explain what is sought beyond direct iteration on the finite field.

Short English statement

Given a quadratic polynomial over a finite field and a starting value, determine how many iterations occur before the orbit first enters a cycle and how many distinct elements that cycle contains. The intended form or efficiency of a general determination is unspecified. The supplied registry labels the problem partially solved; this draft does not independently verify that assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003274$, title Preperiods and Cycles of Quadratic Polynomial Iteration, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2016 and names no proposer.
• Supplied workshop provenance: Analytic Number Theory (2017), Oberwolfach Reports, pp. 2975--3030, DOI 10.4171/owr/2016/53; https://doi.org/10.4171/owr/2016/53. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference labeled "Heath-Brown 2017": https://doi.org/10.1112/S0025579317000328.
• Supplied literature reference labeled "Heath-Brown preprint": https://arxiv.org/abs/1701.02707.
• Literature assessment dated 2026-08-21: supplied dated curation derived from the OWR record, not independently verified. It labels the problem partially solved and states that Heath-Brown proved strong cycle-length bounds and analyzed important quadratic families, while the exact cycle length for a general quadratic polynomial over a finite field remains undetermined. The supplied background explicitly requires independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, explaining that mathematical content was preserved while notation, typography, or omitted definitions were repaired for self-contained publication. This is supplied registry metadata, not independent verification of this draft.