S-Integral Points in Orbits on ℙ¹

2023/12/08 by Jit Wu Yap, Yap, Jit Wu
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2312.05094

openalex publication_date 2023/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a number field and S a finite set of places of K that contains all of the archimedean places. Let φ: ℙ¹ → ℙ¹ be a rational map of degree d ≥ 2 defined over K. Given α∈ ℙ¹(K) non-preperiodic and β∈ ℙ¹(K) non-exceptional, we prove an upper bound of the form O(|S|^1+ε) on the number of points in the forward orbit of α that are S-integral relative to β, extending results of Hsia--Silverman [HS11]. We also prove uniform bounds when φ is a polynomial, extending resaults of Krieger et al [KLS+15].

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