Rational and integral values of rational functions at rational points

Abstract: The basic issue concerns sets of values of rational functions at rational points of an algebraic variety, namely image f(X(k)), where X is an algebbaric variety and f is a rational function on X, defined over the number field k. For instance, we shall prove that for X an abelian variety, the map between rational points is never surjective. This is reminiscent of the Hilbert Property, but here the fibers may have arbitrary dimension. One of our examples concerns the classical Hilbert Property: we produce a simply connected affine surface whose set of integral points is Zariski-dense and thin, disproving a plausible expectation. We shall also discuss hieghts and integrality issues; in this context, a role will be played by 'gcd estimates'. In the first Appendix, written by D. Masser, an effective estimate of some relevant gcd is provided.