A Stothers–Mason theorem with a difference radical

Abstract

Differential calculus is not a unique way to observe polynomial equations such as $$a+b=c$$ 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.

Difference radical in terms of shifting zero and applications to the Stothers-Mason theorem

“…On the other hand, the ideas in [4] work for considering some problems even though the definitions are different. Actually, there are several properties in common.…”

“…The equation (4.4) was considered in [4,Theorem 3.5] by the estimates of degree of Casoratian C(z) replacing ∆ k f i with f i (z + k), i = 1, 2, . .…”

“…We have proved Theorem 4.1. 4 , which are are linearly independent over C, and consider a polynomial…”

“…We study a difference analogue of radical of a complex polynomial, and consider a difference analogue of the Stothers-Mason theorem. As applications, we study the difference version of the Fermat type functional equations, see e.g., [4], [9].…”

“…Let a, b and c be relatively prime polynomials such that not all of them are identically zero. The Stothers-Mason theorem [10], [16], see also e.g., [15] A difference analogue of the Stothers-Mason theorem is seen in e.g., [4]. One of the purpose of this paper is that we give an alternative approach to difference analogues of radical of polynomials in terms of shifting zero different from [4].…”