On the Equations Z

We investigate integer solutions of the superelliptic equation (1) z = F (x, y), where F is a homogenous polynomial with integer coefficients, and of the generalized Fermat equation (2) Ax + By = Cz, where A,B and C are non-zero integers. Call an integer solution (x, y, z) to such an equation proper if gcd(x, y, z) = 1. Using Faltings’ Theorem, we shall show that, other than in certain exceptional circumstances, these equations have only finitely many proper solutions. We examine (1) using a descent technique of Kummer, which allows us to obtain, from any infinite set of proper solutions to (1), infinitely many rational points on a curve of (usually) high genus, thus contradicting Faltings’ Theorem (for example, this works if F (t, 1) = 0 has three simple roots and m ≥ 4). We study (2) via a descent method which uses unramified coverings of P1−{0, 1,∞} of signature (p, q, r). ¿From infinitely many proper solutions to (2) we obtain infinitely many rational points on some curve of (usually) high genus in some number field, thus contradicting Faltings’ Theorem if 1/p + 1/q + 1/r < 1. We then collect together a variety of results for (2) when 1/p+ 1/q +1/r ≥ 1. In particular we consider ‘local-global’ principles for proper solutions, and consider solutions in function fields.

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