An algorithm for Runge's method

Let F(x,y) be an absolutely irreducible polynomial in two
variables with integer coefficients. The diophantine equation
F(x,y)=0 in integers x,y can be solved explicitly for very
limited classes of polynomials F. We give a brief discussion
of the results in this direction. One of those results was obtained
by Runge in 1887 who showed that under the so-called Runge-condition
the equation F(x,y)=0 can be solved completely. In this lecture we show
how Runge's idea can be turned into an algorithm that can actually
be fed to a computer. This work is carried out jointly with S.Tengely.