2016/11/13 by Liang-Chung Hsia, Hsia, Liang-Chung, Thomas J. Tucker +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1611.04115
Following work of Bugeaud, Corvaja, and Zannier for integers, Ailon and Rudnick prove that for any multiplicatively independent polynomials, a, b ∈ \mathbb C[x], there is a polynomial h such that for all n, we have gcd(an - 1, bn - 1) | h We prove a compositional analog of this theorem, namely that if f, g ∈ \mathbb C[x] are nonconstant compositionally independent polynomials and c(x) ∈ \mathbb C[x], then there are at most finitely many λ with the property that there is an n such that (x - λ) divides gcd(f∘ n(x) - c(x), g∘ n(x) - c(x)).