Open Problems in Arithmetic Dynamics — Question 20

Definitions and mathematical statement

The supplied record uses the symbol $\operatorname{ShafDim}_d^N$ without defining it or specifying the admissible indices $d,N$. Retaining the notation of the cited source, the question is to determine the exact value of \[ \operatorname{ShafDim}_d^N \] for all admissible pairs $(d,N)$. In particular, the source asks whether \[ \operatorname{ShafDim}_d^1=d+1\qquad\text{for all }d\geq 3. \] The index domains, the definition of the invariant, and any hypotheses entering that definition must be recovered from the cited source before this can be a self-contained mathematical statement. The displayed assertion retains the source's bound and quantifier without supplying an unstated index domain.

Short English statement

Determine the invariant $\operatorname{ShafDim}_d^N$, and in particular decide whether $\operatorname{ShafDim}_d^1=d+1$ for every $d\geq3$. The supplied record omits the definition and index domains, so the formulation remains incomplete.

Sources

• Local UnsolvedMath registry, permanent ID $800397$, source problem code catalogue entry, title Open Problems in Arithmetic Dynamics --- Question 20. The supplied registry labels the problem open; this status has not been independently verified.
• Benedetto, DeMarco, Ingram, Jones et al., Current Trends and Open Problems in Arithmetic Dynamics (2018), Question 20. Supplied source URL: https://arxiv.org/abs/1806.04980. The registry attributes the problem to these authors and gives proposed year 2018.
• Supplied provenance records access on 2026-10-16 and extraction method arxiv_latex. These are supplied provenance claims, not a claim of independent source access or verification for this draft.