A classical tool for studying Hilbert's irreducibility theorem is Siegel's finiteness theorem for S-integral points on algebraic curves. We present a different approach based on s-integral points rather than S-integral points. Given an integer s > 0, an element t of a field K is said to be s-integral if the set of places v is an element of M(K) for which \t\(v) > 1 is of cardinality less than or equal to s (instead of contained in S for ''S-integral''). We prove a general diophantine result for s-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 that K is of characteristic p > 0, will be given in forthcoming papers ([8],[9]).
