A Stothers\u2013Mason theorem with a difference radical

Katsuya Ishizaki,Nan Li,Kazuya Tohge,Risto Korhonen

doi:10.1007/s00209-020-02604-7

Abstract

Differential calculus is not a unique way to observe polynomial equations such as a+b=c. We propose a way of applying difference calculus to estimate multiplicities of the roots of the polynomials a, b and c satisfying the equation above. Then a difference abc theorem for polynomials is proved using a new notion of a radical of a polynomial. Results, for example, on the non-existence of polynomial solutions to difference Fermat and difference Super-Fermat functional equations are given as applications. We also introduce a truncated second main theorem for differences, and use it to consider these functional equations with non-polynomial entire solutions. Equations with polynomial or non-polynomial solutions are observed to see the sharpness of results obtained.

Highlights

The Stothers–Mason theorem states that if relatively prime polynomials a, b and c, not all of them identically zero, satisfy a + b = c, deg c ≤ deg rad(abc) − 1, where the radical rad(abc) is the product of distinct linear factors of abc [27,37], see [13,35]
Has a solution consisting of m polynomials f1, f2, . . . , fm, n ≤ m2 −m −1
The Stothers–Mason theorem has been generalized in many different directions, for instance, to sums in one-dimensional function fields by Mason [28], by Voloch [43] and by Brownawell and Masser [2], to sums of pairwise relatively prime polynomials of several variables by Shapiro and Sparer [32], to sums in higher-dimensional function fields by Hsia and Wang [19], and to quantum deformations of polynomials by Vaserstein [41]

Summary

Introduction

The Stothers–Mason theorem states that if relatively prime polynomials a, b and c, not all of them identically zero, satisfy a + b = c, deg c ≤ deg rad(abc) − 1, where the radical rad(abc) is the product of distinct linear factors of abc [27,37], see [13,35]. The purpose of this paper is to introduce a difference counterpart of the radical, and to use it to prove a difference analogue of the Stothers–Mason theorem, as well as a truncated version of the difference second main theorem for holomorphic curves.

Difference radical

Difference analogue of the Stothers–Mason theorem

Polynomial solutions of difference Super-Fermat functional equations

Discussion