Generalized Greatest Common Divisors for the Orbits under Rational Functions

Abstract

Assume Vojta's Conjecture. Suppose $a,b,\alpha ,\beta \in \mathbb{Z}$, and $f(x),g(x)\in \mathbb{Z}[x]$ are polynomials of degree $d\ge 2$. Assume that the sequence $(f^{\circ n}(a),g^{\circ n}(b){)}_n$ is generic and $\alpha ,\beta$ are not exceptional for $f,g$ respectively, we prove that for each given $\epsilon >0$, there exists constant $C=C(\epsilon ,a,b,\alpha ,\beta ,f,g)>0$, such that for all $n\ge 1$, we have $$\gcd (f^{\circ n}(a)-\alpha ,g^{\circ n}(b)-\beta )\le C\cdot \exp (\epsilon \cdot d^n).$$ We prove an estimate for rational functions and for a more general gcd and then obtain the above inequality as a consequence.

arXiv comment18 pages