2026/06/30 by Brian Kintu
Mathematics · #math.NT #math.DS
31 pages, as also my sincerest congratulations to Prof. Jacob Tsimerman, and any comments are very welcome!
arxiv created 2026/08/01 · arxiv updated 2026/08/04
In this follow-up article of a multi-part series on (strictly) preperiodic point-counting, we inspect an astonishing relationship between the set of (strictly) 1m-preperiodic points of a polynomial map φd, c defined by φd, c(z) = zd + c for all c, z ∈ OK and the coefficient c, where K is any number field of degree n≥ 1, d>2 is an integer and m∈ ℤ≥ 1 is any fixed (eventual period). As before, we wish to study counting problems that are inspired by torsion point-counting in arithmetic statistics and (strictly) preperiodic point-counting in arithmetic dynamics. In doing so, we then first prove that for any prime p≥ 3 and for any fixed ℓ ∈ ℤ≥ 1 and fixed (eventual period) m∈ ℤ≥ 1, the average number of distinct 1m-preperiodic integral points of any odd degree map φpℓ, c modulo prime ideal pOK is unbounded or zero as c tends to infinity. Inspired further by work of Doyle-Poonen, along with conjectural work of Hutz and abc(d)-conditional work of Panraksa on K-rational preperiodic points of any even degree map φ(p-1)ℓ, c for any prime p≥ 5 in arithmetic dynamics, we then also prove that for any fixed (eventual period) m ∈ ℤ ≥ 1, the average number of distinct 1m-preperiodic integral points of any φ(p-1)ℓ, c modulo prime ideal pOK is unbounded or zero as c→ ∞. Finally, we then apply density, polynomial- and number field-counting, and Sato-Tate equidistribution results from arithmetic statistics, and thereby obtaining further a stream of counting and statistical results on arithmetic objects that arise naturally in our polynomial discrete dynamical settings.