Generalized greatest common divisors for orbits under rational functions
K Huang - Monatshefte für Mathematik, 2020 - Springer
K Huang
Monatshefte für Mathematik, 2020•SpringerAbstract Assume Vojta's Conjecture for blowups of P^ 1 * P^ 1 P 1× P 1. Suppose a, b, α, β
∈ Z a, b, α, β∈ Z, and f (x), g (x) ∈ Z xf (x), g (x)∈ Z x are polynomials of degree d ≥ 2 d≥
2. Assume that the sequence (f^ ∘ n (a), g^ ∘ n (b)) _n (f∘ n (a), g∘ n (b)) n is generic and α,
β α, β are not exceptional for f, g respectively. We prove that for each given ε> 0 ε> 0, there
exists a constant C= C (ε, a, b, α, β, f, g)> 0 C= C (ε, a, b, α, β, f, g)> 0, such that for all n ≥ 1
n≥ 1, we have (f^ ∘ n (a)-α, g^ ∘ n (b)-β) ≤ C ⋅\exp (ε ⋅ d^ n). gcd (f∘ n (a)-α, g∘ n (b)-β)≤ …
∈ Z a, b, α, β∈ Z, and f (x), g (x) ∈ Z xf (x), g (x)∈ Z x are polynomials of degree d ≥ 2 d≥
2. Assume that the sequence (f^ ∘ n (a), g^ ∘ n (b)) _n (f∘ n (a), g∘ n (b)) n is generic and α,
β α, β are not exceptional for f, g respectively. We prove that for each given ε> 0 ε> 0, there
exists a constant C= C (ε, a, b, α, β, f, g)> 0 C= C (ε, a, b, α, β, f, g)> 0, such that for all n ≥ 1
n≥ 1, we have (f^ ∘ n (a)-α, g^ ∘ n (b)-β) ≤ C ⋅\exp (ε ⋅ d^ n). gcd (f∘ n (a)-α, g∘ n (b)-β)≤ …
Abstract
Assume Vojta’s Conjecture for blowups of . Suppose , and are polynomials of degree . Assume that the sequence is generic and are not exceptional for f, g respectively. We prove that for each given , there exists a constant , such that for all , we have $$\begin{aligned} \gcd (f^{\circ n}(a)-\alpha , g^{\circ n}(b) -\beta ) \le C\cdot \exp ({\varepsilon \cdot d^n}). \end{aligned}$$We prove an estimate for rational functions and for a more general gcd and then obtain the above inequality as a consequence.
Springer
以上显示的是最相近的搜索结果。 查看全部搜索结果