Generalized Greatest Common Divisors for the Orbits under Rational Functions

2017/02/13 by Huang, Keping · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1702.03881

Abstract

Assume Vojta's Conjecture. Suppose a, b, α,β∈ ℤ, and f(x),g(x) ∈ ℤ[x] are polynomials of degree d ≥ 2. Assume that the sequence (f^{∘ n}(a), g^{∘ n}(b))_n is generic and α,β are not exceptional for f,g respectively, we prove that for each given ε > 0, there exists constant C = C(ε,a,b,α,β,f,g)>0, such that for all n ≥ 1, we have gcd(f^{∘ n}(a)-α, g^{∘ n}(b) -β) ≤ C⋅exp(ε⋅ dⁿ). We prove an estimate for rational functions and for a more general gcd and then obtain the above inequality as a consequence.

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