Hilbert subsets ands-integral points

Abstract

A classical tool for studying Hilbert's irreducibility theorem is Siegel's finiteness theorem forS-integral points on algebraic curves. We present a different approach based ons-integral points rather thanS-integral points. Given an integers>0, an elementt of a fieldK is said to bes-integral if the set of placesv ∈M K for which |t|v > l is of cardinality ≤s (instead of contained inS for “S-integral”). We prove a general diophantine result fors-integral points (Th.1.4). This result, unlike Siegel's theorem, is effective and is valid more generally for fields with the product formula. The main application to Hilbert's irreducibility theorem is a general criterion for a given Hilbert subset to contain values of given rational functions (Th.2.1). This criterion gives rise to very concrete applications: several examples are given (§2.5). Taking advantage of the effectiveness of our method, we can also produce elements of a given Hilbert subset of a number field with explicitely bounded height (Cor.3.7). Other applications, including the case thatK is of characteristicp>0, will be given in forthcoming papers ([8], [9]).