2024/12/30 by Brian Kintu, Kintu, Brian · 4 citations
Mathematics · #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2501.04026
openalex publication_date 2024/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this first article of a multi-part series, we inspect a surprising relationship between the set of fixed points of a polynomial map φd, c defined by φd, c(z) = zd + c for all c, z ∈ ℤ and the coefficient c, where d > 2 is an integer. Inspired greatly by the elegant counting problems along with the very striking results of Bhargava-Shankar-Tsimerman and their collaborators in arithmetic statistics, and also by interesting point-counting result of Narkiewicz on rational periodic points of any odd degree map φd, c in arithmetic dynamics, we then first prove that for any prime p≥ 3, the average number of distinct integral fixed points of any φp, c modulo p is 3 or 0 as c tends to infinity. Inspired further by a conjecture of Hutz on rational periodic points of φp-1, c for any prime p≥ 5 in arithmetic dynamics, we then also prove that the average number of distinct integral fixed points of any φp-1, c modulo p is 1 or 2 or 0 as c→ ∞. Finally, we then apply density and number field-counting results from arithmetic statistics, and as a result obtain counting and statistical results on the irreducible integer polynomials and number fields arising naturally in our polynomial discrete dynamical settings.