Estimates for the solutions fo certain diophantine equations bt runge's method

Abstract

We consider the two Diophantine equations ym = F(x) and G(y) = F(x) under the assumption that gcd (m, deg F) > 1 and gcd (deg G, deg F) > 1, respectively. We prove that the bounds for the denominator of the coefficients of the power series arising from the above two situations can be improved considerably and thus we establish improved upper bounds for the size of the solutions (namely for x and y ). We also give explicit upper bounds for the integer solutions of equations of the form F(x,y) = P1(x) Q2(y) - P2(y) Q1(x) = under the assumption that gcd(deg, P1 - deg Q1,deg P2 - deg Q2) > 1. © 2008 World Scientific Publishing Company.