Asymptotic gcd and divisible sequences for entire functions

Abstract:

Let $f$ and $g$ be algebraically independent entire functions. We first give an estimate of the Nevanlinna counting function for the common zeros of $f^n-1$ and $g^n-1$ for sufficiently large $n$. We then apply this estimate to study divisible sequences in the sense that $f^n-1$ is divisible by $g^n-1$ for infinitely many $n$. For the first part of establishing our gcd estimate, we need to formulate a truncated second main theorem for effective divisors by modifying a theorem from a paper by Hussein and Ru and explicitly computing the constants involved for a blowup of $\mathbb {P}^1\times \mathbb {P}^1$ along a point with its canonical divisor and the pull-back of vertical and horizontal divisors of $\mathbb {P}^1\times \mathbb {P}^1$.