A fast algorithm for finding small solutions of F(X, Y) = G(X, Y) over number fields

Let M be a number field of degree m with ring of integers Z(M). Let F is an element of Z(M) [X, Y] be a form of degree n such that F(X,1) has distinct roots. Let G is an element of Z[X, Y] be an arbitrary polynomial of degree k. Assuming that k <= n-2m+1 if all roots of F-(i)(X, 1) (1 <= i <= n) are complex and k <= n-4m+1 otherwise, we provide an efficient algorithm for finding all solutions X, Y is an element of Z(M), max (vertical bar X vertical bar, vertical bar Y vertical bar) < C of the inequality vertical bar F(X,Y)vertical bar <= c (.) vertical bar G(X,Y)vertical bar. We provide numerical examples with m = 3 and C = 10(100).