EFFECTIVE ANALYSIS OF INTEGRAL POINTS ON ALGEBRAIC-CURVES

Let K be an algebraic number field, S superset of or equal to S infinity a finite set of valuations and C a non-singular algebraic curve over K. Let x is an element of K(C) be non-constant. A point P is an element of C(K) is S-integral if it is not a pole of x and \x(P)\ upsilon > 1 implies upsilon is an element of S. It is proved that all S-integral points can be effectively determined if the pair (C, x) satisfies certain conditions. In particular, this is the case if (i) x: C --> P-1 is a Galois covering and g(C) greater than or equal to 1; (ii) the integral closure of ($) over bar Q[x] in ($) over bar Q(C) has at least two units multiplicatively independent mod ($) over bar Q*. This generalizes famous results of A. Baker and other authors on the effective solution of Diophantine equations.