Open Problems in Arithmetic Dynamics — Question 16

Definitions and mathematical statement

Retain the source notation $\operatorname{FOM/FOD}(N,d)$. The supplied record does not define this quantity, its underlying objects, or its field hypotheses; consequently, the question cannot yet be made self-contained. The integer domains for the indices used below are a provisional interpretation requiring confirmation from the cited source.

Under that interpretation, for each $N\in{\mathbb{Z}}_{\ge1}$, set formally \[ B(N):=\sup_{d\in{\mathbb{Z}}_{\ge2}}\operatorname{FOM/FOD}(N,d). \] The question is whether \[ B(N)<\infty\qquad\text{for every }N\in{\mathbb{Z}}_{\ge1}. \] If this holds, determine the exact value of $B(N)$ as a function of $N$.

The supplied motivation states that a bound for $\operatorname{FOM/FOD}(1,d)$ is independent of $d$. Neither the definition of this quantity nor the value and hypotheses of that bound are included in the record. This motivation is retained as a source assertion, not an independently verified result.

Short English statement

For each fixed $N$, ask whether the source quantity $\operatorname{FOM/FOD}(N,d)$ is uniformly bounded as $d\ge2$ varies, and, if so, determine its supremum as a function of $N$. The quantity and its hypotheses are missing from the supplied excerpt, and the integer indexing is provisional.

Sources

• Local UnsolvedMath registry, permanent ID $800393$, source problem code catalogue entry, entitled Open Problems in Arithmetic Dynamics --- Question 16. The supplied registry attributes the question to Benedetto, DeMarco, Ingram, Jones et al. (2018) and labels it open; this status is supplied curation, not an independently verified literature assessment.
• Benedetto, DeMarco, Ingram, Jones et al., Current Trends and Open Problems in Arithmetic Dynamics (2018), Question 16. Supplied source URL: https://arxiv.org/abs/1806.04980.
• Supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are source-record provenance claims; no independent source visit or verification is asserted. Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.