On the Number of Small Points for Rational Maps

2025/10/14 by Jit Wu Yap, Yap, Jit Wu
Mathematics · #Mathematics and Applications #Mathematical Dynamics and Fractals #Advanced Differential Equations and Dynamical Systems

paper · pdf · doi:10.48550/arxiv.2510.12039

Abstract

Let K be a number field and f: ℙ¹ → ℙ¹ a rational map of degree d ≥ 2 with at most s places of bad reduction, where we include all archimedean places. We prove that there exists constants c₁,c₂ > 0, depending only on d and not on f or K, such that # \ x ∈ ℙ¹(K) | \widehath_f(x) ≤ (c₁)/(s) h_ratd(⟨ f ⟩) \ ≤ c₂ s log(s). Here, rat_d is the moduli space of rational maps up to conjugacy, h_ratd is an ample height and ⟨ f ⟩ is the equivalence class associated to f. This gives a uniform version of a theorem of Baker as well as generalizing the results of Benedetto and Looper from polynomials to rational maps. The main tool used is the degeneration of sequences of rational maps by Luo which has been recently formalized by Favre-Gong via Berkovich spaces.

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• Scarcity of finite orbits for rational functions over a number field
• On quadratic rational maps with prescribed good reduction