Reducibility of polynomials and diophantine equations

Abstract

In this final chapter, we deal with a ‘meta’-topic, one that lies ‘over’ the theory of diophantine equations, as it were; namely arithmetic specialisation of polynomials. The main result asserts that under such specialisations the multiplicative structure of the numbers obtained goes some considerable way towards determining the multiplicative structure of the original polynomials. This allows one to give effective versions of Hilbert's irreducibility theorem and to describe all abelian points on algebraic curves. The methods used are quite independent of the theory of linear forms in the logarithms of algebraic numbers. and rely on the study of the arithmetic structure of sums of algebraic power series in all metrics of the field of rational numbers.