Open Problems in Arithmetic Dynamics — Conjecture 9

Definitions and mathematical statement

Let $K$ be a field of characteristic $0$, let $\bar K$ be an algebraic closure of $K$, and let $f:\mathbb P^N_K\to\mathbb P^N_K$ be a morphism defined over $K$ of degree $d\ge 2$. The source does not explicitly specify the permitted range of the projective dimension $N$.

For a positive integer $r$, define the Möbius function by \[ \mu(r)=\begin{cases} 1,&r=1,\\ (-1)^k,&r\text{ is a product of }k\text{ distinct primes},\\ 0,&r\text{ is divisible by the square of a prime}. \end{cases} \] The supplied extraction of Conjecture 9 states that there exists a positive integer $n_0$, with no uniformity in the given data specified, such that for every integer $n\ge n_0$, \[ \#\operatorname{Per}_n^*\left(f,\mathbb P^N(\bar K)\right) :=\sum_{\substack{m\mid n\\m\ge 1}}\mu\!\left(\frac{n}{m}\right)(d^m+1)^N. \] This reproduces the extracted relation, including its definition symbol $:=$. The supplied record does not define $\operatorname{Per}_n^*$, specify whether its count includes multiplicities, or explain whether the displayed relation is intended as a definition or a conjectural equality. Consequently, the intended mathematical assertion cannot be made unambiguous from this record alone. In particular, neither an interpretation as points of exact period $n$ nor any additional assertion of positivity or nonemptiness is supplied here.

Short English statement

For a morphism of projective space of degree at least $2$ over a field of characteristic $0$, the extracted conjecture gives an eventual relation between an undefined starred periodic-point count and a Möbius sum. The meaning of the starred count and the intended role of the definition symbol remain unresolved.

Sources

• Local UnsolvedMath registry, supplied source record: permanent ID $800386$, exact source problem code catalogue entry; title: Open Problems in Arithmetic Dynamics --- Conjecture 9. The registry labels the problem open; this is supplied status metadata, not an independently verified mathematical assessment.
• Benedetto, DeMarco, Ingram, Jones et al., Current Trends and Open Problems in Arithmetic Dynamics (2018), Conjecture 9. Supplied source URL: https://arxiv.org/abs/1806.04980. The registry attributes the proposal to these authors in 2018.
• Supplied provenance records access on 2026-10-16 and extraction method arxiv_latex. This access date and provenance are supplied curation information; no independent source visit or verification is claimed.