The Fermat equation with polynomial values as base variables (Q795861)

Let \(n\in\mathbb Z\), \(n>2\). Let \(E,F,G\in \mathbb Z[X]\), non-constant and relatively prime. It is well known that \(E^ n+F^n\neq G^n\). We prove the following refinement. All solutions of the equation \((E(t))^n+(F(t))^n=(G(t))^n\) in rational integers \(n,t\) with \(E(t)F(t)G(t)\neq 0\) satisfy \(\max(n,| t|)<C\) where \(C\) is a computable number depending only on \(E, F\) and \(G\). Similar, but less elegant results are proved for the equations \((E(t))^n+(F(t))^n=z^n\) in rational integers \(n, t, z\) and \((E(t,u))^n+(F(t,u))^n=(G(t,u))^n\) in rational integers \(n, t, u\) where in the latter case \(E, F\) and \(G\) are homogeneous polynomials in two variables. The proofs are based on estimates for linear forms in logarithms of algebraic numbers. In particular, we apply a generalization of results of Sprindzhuk and Turk by giving an upper bound for the exponent \(m\) in the equation \(a f(x)=bw y^m\) where the fixed \(f\in \mathbb Z[X]\) has at least two distinct zeros and \(a,b,m,w,x\) and \(y\) are rational integers with \(ab\neq 0\), \(| y|>1\), \(a\) and \(b\) fixed and \(w\) composed of a finite set of fixed prime.