Another generalization of Mason's ABC-theorem

TL;DR

This paper extends Mason's ABC-theorem to multivariate polynomials, providing new conditions under which certain polynomial equations, including generalized Fermat-Catalan equations, have no non-trivial solutions.

Contribution

It introduces a broader version of Mason's ABC-theorem applicable to multivariate polynomials with specific gcd and subsum conditions, and applies it to polynomial Fermat-type equations.

Findings

No non-constant solutions for certain polynomial Fermat-Catalan equations under specified conditions.

No 'interesting' solutions for polynomial equations with equal exponents when the exponent exceeds a threshold.

Generalization of Mason's ABC-theorem to multivariate polynomials with gcd and subsum constraints.

Abstract

We show a generalization of Mason's ABC-theorem, with the only conditions that the greatest common divisor has been divided out and no proper subsum of the (possibly multivariate) polynomial sum f_1 + f_2 + ... + f_n = 0 vanishes. As a result, we show that the generalized Fermat-Catalan equation for polynomials: g_1^{d_1} + g_1^{d_2} + ... + g_n^{d_n} = 0 has no non-constant solutions if the greatest common divisor of the terms equals one, no proper subsum vanishes and the hyperbolic sum 1/d_1 + 1/d_2 + ... + 1/d_n is at most 1/(n-2). Furthermore, we show that the generalized Fermat-equation for polynomials g_1^d + g_1^d + ... + g_n^d = 0 has no 'interesting' solutions if d >= n(n-2).