Notes on a generalized abc-conjecture over function fields

Pei-Chu Hu,Chung-Chun Yang

doi:10.5802/ambp.135

Abstract

In this note, we weaken a condition in the generalized abo-conjecture proposed by us in a previous paper, and prove its analogue for non-Archimedean entire functions, as well as a generalized Mason's theorem for polynomials.

Highlights

In all the paper, x will denote an algebraically closed field of characteristic zero.Let a be a non-zero integer
Under a stronger condition that fo, f~ have no common zeros for j =1, ..., k, some special cases of Theorem 1.1 were given in [7], [9]
Define the maximum term: (r,f) = max n~0|an|rn with the associated the central index: (r, 1 f) n just is the counting function of zeros of /, which denotes the number of zeros of f with absolute value::; r

Summary

Introduction

X will denote an algebraically closed field of characteristic zero. Under a stronger condition that fo, f~ have no common zeros for j =1, ..., k, some special cases of Theorem 1.1 were given in [7], [9]. Theorem 1.2 yields immediately the following: Theorem 1.3 Let fl, f 2, ’ ’ ’ , fk (k > 2) be linearly independent polynomials in It. Put f o = f i + f 2 + fk + ... Theorem 1.4 For fixed integer k ~ 1, let fj (j = k) 0, ..., be non-zero polynomials on r~ such that f i + ~ + ~ f k = f o. As an application of Theorem 1.2, we can derive the following: Theorem 1.5 Given polynomials f1, f2, ..., fk (k ~ 2) in K, and positive integers h(1 j k) such that (a) fl, fl22, ,... Davenport proved that (9) is true as long as f 2 g3 -- ~ 0 ( see ~1~,(18~)

Basic facts

Proof of the main theorems