
Asymptotic gcd and divisible sequences for entire functions
Author(s) -
Guo Ji-dong,
Julie Tzu-Yueh Wang
Publication year - 2019
Publication title -
transactions of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.798
H-Index - 100
eISSN - 1088-6850
pISSN - 0002-9947
DOI - 10.1090/tran/7435
Subject(s) - algorithm , annotation , computer science , type (biology) , artificial intelligence , database , mathematics , biology , ecology
Let f f and g g be algebraically independent entire functions. We first give an estimate of the Nevanlinna counting function for the common zeros of f n − 1 f^n-1 and g n − 1 g^n-1 for sufficiently large n n . We then apply this estimate to study divisible sequences in the sense that f n − 1 f^n-1 is divisible by g n − 1 g^n-1 for infinitely many n 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 P 1 × P 1 \mathbb {P}^1\times \mathbb {P}^1 along a point with its canonical divisor and the pull-back of vertical and horizontal divisors of P 1 × P 1 \mathbb {P}^1\times \mathbb {P}^1 .