2025/03/14 by Brian Kintu, Kintu, Brian
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2503.11393
openalex publication_date 2025/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this follow-up paper, we again inspect a surprising connection between the set of fixed 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 and d > 2 is an integer. As before, we wish to study counting problems which are inspired by exciting advances in arithmetic statistics, and again partly by point-counting result of Narkiewicz on real K-rational periodic points of any odd degree map φd,c in arithmetic dynamics. In doing so, we then first prove that for any real algebraic number field K of degree n ≥ 2, and for any prime p ≥ 3 and integer ℓ ≥ 1, the average number of distinct integral fixed points of any φpℓ,c modulo prime ideal pOK is 3 or 0 as c→ ∞. Motivated further by K-rational periodic point-counting result of Benedetto on any φ(p-1)ℓ,c for any prime p ≥ 5 and integer ℓ ∈ ℤ≥ 1 in arithmetic dynamics, we then also prove unconditionally that for any number field (not necessarily real) K of degree n ≥ 2, the average number of distinct integral fixed points of any φ(p-1)ℓ,c modulo prime pOK 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 irreducible polynomials and number fields arising naturally in our polynomial discrete dynamical settings.