A fast algorithm for finding "small" solutions of F(x,y)=G(x,y) over imaginary quadratic fields

Let d be a square-free positive integer, and K = ( ) an imaginary quadratic field. Let F(X,Y), G(X,Y) ε K[X,Y] be polynomials of degree n and m , respectively. Assume, that F is homogeneous, irreducible and n - m ≥ 4. We give a fast algorithm (based on the LLL basis reduction algorithm) for finding the solutions of the inequality F(X,Y) ≤ G(X,Y) in X, Y ε kX,y