Mason's theorem with a difference radical

TL;DR

This paper introduces a novel approach using difference calculus and a new radical concept to analyze polynomial roots and proves a difference version of Mason's theorem, with applications to difference Fermat equations.

Contribution

It presents a new difference radical notion and establishes a Mason's theorem analogue for polynomials, extending to transcendental solutions.

Findings

Proved a difference Mason's theorem for polynomials

Established non-existence results for difference Fermat equations

Introduced a truncated second main theorem for differences

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. Two results on the non-existence of polynomial solutions to difference Fermat type functional equations are given as applications. We also introduce a truncated second main theorem for differences, and use it to consider difference Fermat type equations with transcendental entire solutions.