2024/07/29 by Alina Ostafe, Ostafe, Alina, Igor E. Shparlinski +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Advanced Mathematical Theories and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2407.20464
openalex publication_date 2024/07/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Towards a well-known open question in arithmetic dynamics, L. Mérai, A. Ostafe and I. E. Shparlinski (2023), have shown, for a class of polynomials f ∈ \mathbb Z[X], which in particular includes all quadratic polynomials, that, under some natural conditions (necessary for quadratic polynomials), the set of primes p, such that all iterations of f are irreducible modulo p, is of relative density zero, with an explicit estimate on the rate of decay. This result relies on some bounds on character sums via the Brun sieve. Here we use the Selberg sieve and in some cases obtain a substantial quantitative improvement.