Open Problems in Arithmetic Dynamics — Conjecture 8 (Odoni, 1985, [MR805714])

Definitions and mathematical statement

For a field $K$, write ${\mathbb{P}}^1(K)$ for its projective line of $K$-rational points. The supplied record calls $K$ Hilbertian if there do not exist finitely many smooth curves $C_i/K$ and finite maps $F_i:C_i\to{\mathbb{P}}^1_K$, each of degree at least $2$, such that \[ {\mathbb{P}}^1(K)=\bigcup_{i=1}^{r}F_i\left(C_i(K)\right). \] The record gives all number fields as examples and states that ${\mathbb{R}}$, ${\mathbb{C}}$, and ${\mathbb{Q}}_p$ are not Hilbertian. The conventions concerning the curves and maps in this characterization are not further specified in the supplied record.

The conjecture attributed to Odoni (1985), reproduced as Conjecture 8 in the supplied 2018 source, is the following: for every Hilbertian field $K$ and every integer $d\geq2$, there exist coefficients $a_0,\ldots,a_{d-1}\in K$ and a point $P\in K$ such that the monic polynomial \[ f(X)=X^d+a_{d-1}X^{d-1}+\cdots+a_1X+a_0 \] satisfies \[ \iota_K(f,P)=1. \] The notation $\iota_K(f,P)$ is retained exactly from the source record, which does not define it. Consequently, the displayed condition is an unresolved part of the formulation; this draft does not assign it a meaning.

The supplied addendum, attributed to Jones (2015), asserts that when $K$ is a number field, the polynomial can additionally be chosen in $\mathcal O_K[X]$, where \[ \mathcal O_K=\{\alpha\in K:\alpha\text{ satisfies a monic polynomial in }{\mathbb{Z}}[X]\}. \] The requirements $\deg f=d$, monicity, $P\in K$, and $\iota_K(f,P)=1$ are retained in this addendum. Its attribution is preserved without independently assessing its literature status.

Short English statement

For every Hilbertian field $K$ and integer $d\geq2$, the source conjectures the existence of a monic degree-$d$ polynomial $f\in K[X]$ and $P\in K$ satisfying $\iota_K(f,P)=1$. The source does not define $\iota_K$, so this condition still requires clarification. A supplied addendum allows integral coefficients when $K$ is a number field.

Sources

• Local UnsolvedMath registry: permanent ID $800385$, exact source problem code catalogue entry. The supplied record labels the problem open; this label is source metadata, not an independently verified mathematical assessment.
• Benedetto, DeMarco, Ingram, Jones et al., Current Trends and Open Problems in Arithmetic Dynamics (2018), Conjecture 8. Supplied source URL: https://arxiv.org/abs/1806.04980. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent access or verification.
• Original conjecture attribution supplied by the record: Odoni (1985), reference label MR805714. No further bibliographic details are supplied.
• Number-field addendum attribution supplied by the record: Jones (2015), reference label MR3220023. No further bibliographic details are supplied.