Open Problems in Arithmetic Dynamics — Question 14

Definitions and mathematical statement

Let $c$ be an indeterminate, let ${\mathbb{Q}}[c]$ be the polynomial ring over the rational numbers, and consider the family $f_{d,c}(x)=x^d+c$. The supplied record denotes by $G_{d,m,n}(c)\in{\mathbb{Z}}[c]$ the associated "Gleason polynomial." It gives no defining formula, normalization, or interpretation of the indices $m,n$; consequently, this designation must remain source-dependent in this draft.

A polynomial $P\in{\mathbb{Q}}[c]$ is irreducible if it is nonzero, is not a unit, and every factorization $P=AB$ in ${\mathbb{Q}}[c]$ has at least one constant factor. For a nonzero polynomial, a complete factorization over ${\mathbb{Q}}$ means an expression \[ P(c)=a\prod_{i=1}^{r}p_i(c)^{e_i}, \] where $a\in{\mathbb{Q}}\setminus\{0\}$, the $p_i$ are distinct monic irreducible polynomials in ${\mathbb{Q}}[c]$, and each $e_i$ is a positive integer. The empty product covers nonzero constant polynomials.

The supplied questions are:

1. (a) Is $G_{2,m,n}(c)$ irreducible in ${\mathbb{Q}}[c]$ for every pair of integers $m\geq 0$ and $n\geq 1$?
2. (b) For which triples $(d,m,n)$ is $G_{d,m,n}(c)$ irreducible in ${\mathbb{Q}}[c]$?
3. (c) Determine the complete factorization of $G_{d,m,n}(c)$ in ${\mathbb{Q}}[c]$.

The admissible triples in parts (b) and (c) are not explicitly specified in the supplied record. In particular, the domain of $d$, the continuation of the bounds on $m,n$, and any exceptional index conventions require clarification. The missing definition of $G_{d,m,n}$ also prevents determining whether zero or constant polynomials occur; under the irreducibility convention above, such polynomials are not irreducible.

Short English statement

The question asks whether every quadratic Gleason polynomial in the stated index range is irreducible over ${\mathbb{Q}}$, which general members are irreducible, and how all members factor over ${\mathbb{Q}}$. A precise standalone formulation requires the missing definition of the Gleason polynomials and the admissible general index range.

Sources

• Local UnsolvedMath registry: permanent ID $800391$, exact source problem code catalogue entry, title "Open Problems in Arithmetic Dynamics --- Question 14." The registry attributes the problem to Benedetto, DeMarco, Ingram, Jones et al., with proposed year 2018.
• Supplied reference: Benedetto, DeMarco, Ingram, Jones et al., Current Trends and Open Problems in Arithmetic Dynamics (2018), Question 14, https://arxiv.org/abs/1806.04980. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not an independently verified source visit.
• The registry labels the problem "open." This is a supplied status label, not an independently verified mathematical or literature assessment. The supplied upstream revision is d371ee917ac9647fbb06cd6fc529deec6f3dda82.