TOTALLY REAL THUE INEQUALITIES OVER IMAGINARY QUADRATIC FIELDS

Let F(x, y) be an irreducible binary form of degree >= 3 with integer coefficients and with real roots. Let M be an imaginary quadratic field with ring of integers Z(M). Let K > 0. We describe an efficient method how to reduce the resolution of the relative Thue inequalities vertical bar F(x, y)vertical bar <= K (x, y is an element of Z(M)) to the resolution of absolute Thue inequalities of type vertical bar F(x,y)vertical bar <= k (x,y is an element of Z). We illustrate our method with an explicit example.