Suppose we are given a rational function with numerator and denominator being polynomials with integer coefficients. Is there an efficient algorithm for finding all integers arguments at which the function takes integer values?

In other words, for given polynomials $F(x)$ and $G(x)$ with integer coefficients, how to compute efficiently all such integers $m$ that $G(m)$ divides $F(m)$ ?

I've developed a rather straight forward approach to this problem at http://list.seqfan.eu/pipermail/seqfan/2010-April/004339.html but I suspect it is far from optimal.