On multiplicative dependence between elements of polynomial orbits

2024/02/21 by Marley Young, Young, Marley
Mathematics · #11D41 #11N25 #37F10 #37P15 #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Graph theory and applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2402.13712

openalex publication_date 2024/02/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We classify the pairs of polynomials f,g ∈ ℂ[X] having orbits satisfying infinitely many multiplicative dependence relations, extending a result of Ghioca, Tucker and Zieve. Moreover, we show that given f₁,…, f_n from a certain class of polynomials with integer coefficients, the vectors of indices (m₁,…,m_n) such that f₁^m₁(0),…,f_n^m_n(0) are multiplictively dependent are sparse. We also classify the pairs f,g ∈ ℚ[X] such that there are infinitely many (x,y) ∈ ℤ² satisfying f(x)^k=g(y)^ℓ for some (possibly varying) non-zero integers k,ℓ.

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