Greatest common divisors of analytic functions and Nevanlinna theory on algebraic tori

Abstract

We study upper bounds for the counting function of common zeros of two meromorphic functions in various contexts. The proofs and results are inspired by recent work involving greatest common divisors in Diophantine approximation, to which we introduce additional techniques to take advantage of the stronger inequalities available in Nevanlinna theory. In particular, we prove a general version of a conjectural "asymptotic gcd" inequality of Pasten and the second author, and consider moving targets versions of our results.

Elliptic Curves with Long Arithmetic Progressions Have Large Rank

“…The result follows 21 We remark that we can use Corollary 6.6 instead of Theorem 1.1 to obtain a version of Theorem 1.6 for more general rational functions, not just x and y-coordinates.…”

“…There are several works in the literature on the problem of improving this trivial bound for the GCD counting function under various assumptions, see for instance [29,23,21]. For our purposes, the following will suffice.…”

A truncated second main theorem for algebraic tori with moving targets and applications

“…GCD in Nevanlinna theory. We recall the gcd counting function of two meromorphic functions and a gcd theorem with moving targets from [12].…”

“…Our proof of Theorem 1.2 relies on the GCD theorem [12] of Levin and the third author and the machinery developed in [9]. The proof of Theorem 1.3 follows the ideas in [13] and [3] with extension to the moving situation.…”

“…, u n are algebraically independent over K.Remark. The proof of[12, Theorem 5.1] actually only treat the case where n ≥ 2 because the case where n = 1 is much easier, as shown in the following argument below. Let u be an entire function without zero.…”