INTEGER POINTS ON ALGEBRAIC-CURVES WITH EXCEPTIONAL UNITS

Let F(X,Y) be an absolutely irreducible polynomial with coefficients i n an algebraic number field K. Denote by C the algebraic curve defined by the equation F(X,Y) = 0 and by K[C] the ring of regular functions on C over K. Assume that there is a unit phi in K[C] - K such that 1 - phi is also a unit. Then we establish an explicit upper bound for the size of integral solutions of the equation F(X. Y) = 0, defined over K. Using this result we establish improved explicit upper bounds on th e size of integral solutions to the equations defining non-singular af fine curves of genus zero, with at least three points at 'infinity', t he elliptic equations and a class of equations containing the Thue cur ves.