Open Problems in Arithmetic Dynamics — Question 30

Definitions and mathematical statement

Let $N\ge 2$ and $d\ge 2$ be integers. The supplied source uses the notation $\overline{\mu}_d(\mathbb{P}^N)$, but the supplied record does not define this quantity or specify the base field of the projective space $\mathbb{P}^N$. Consequently, a self-contained definition of the invariant cannot be recovered from this record.

The source reports the bound \[ \overline{\mu}_d(\mathbb{P}^N)\le d^{-(N-1)} \qquad\text{for all integers }N\ge 2\text{ and }d\ge 2, \] with references [arxiv0908.3835, Proposition 10] and [MR2975153]. This bound is reproduced as a source-reported result.

Question 30. With $\overline{\mu}_d(\mathbb{P}^N)$ understood according to its intended definition in the cited source, determine its exact value for every pair of integers $N\ge 2$ and $d\ge 2$.

Unresolved qualification. The definition of $\overline{\mu}_d$, including its underlying objects, hypotheses, and any quantifiers or extremal operations, must be supplied before this question becomes a precise, self-contained mathematical problem. The stated upper bound alone does not specify that definition or assert equality.

Short English statement

For each pair of integers $N,d\ge 2$, determine the exact value of the source's quantity $\overline{\mu}_d(\mathbb{P}^N)$. The source reports an upper bound of $d^{-(N-1)}$, but the supplied record omits the definition of the quantity and the base field, so the formulation requires supplementation and human mathematical review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800407$, exact source code catalogue entry, title "Open Problems in Arithmetic Dynamics --- Question 30." The registry labels the problem open; this is a supplied status label, not an independently verified literature assessment.
• Benedetto, DeMarco, Ingram, Jones et al., Current Trends and Open Problems in Arithmetic Dynamics (2018), Question 30. Supplied source URL: https://arxiv.org/abs/1806.04980. The record attributes the problem to these authors and gives proposed year 2018.
• The source statement cites [arxiv0908.3835, Proposition 10] and [MR2975153] for the reported upper bound. Full bibliographic details for these references are not supplied.
• Supplied provenance: accessed 2026-10-16; extraction method arxiv_latex; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The access date is supplied provenance; no independent source visit or verification is claimed.