Totally real Thue inequalities over imaginary quadratic fields

Abstract: Let $F(x,y)$ be an irreducible binary form of degree $\geq 3$ 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 [ |F(x,y)|\leq K \;\; (x,y\in Z_M) ] to the resolution of absolute Thue inequalities of type [ |F(x,y)|\leq k \;\; (x,y\in Z). ] We illustrate our method with an explicit example.