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Let $\alpha$ be an algebraic number of degree $d\ge 3$ having at most onereal conjugate and let $K$ be the algebraic number field ${\mathbf Q}(\alpha)$.For any unit $\epsilon$ of $K$ such that ${\mathbf Q}(\alpha\epsilon)=K$, weconsider the irreducible polynomial $f_\epsilon(X)\in{\mathbf Z}[X]$ such that$f_\epsilon(\alpha\epsilon)=0$. Let $F_\epsilon(X,Y)\ =Y^df_\epsilon(X/Y)\in{\mathbf Z}[X,Y]$ be the associated binary form. For eachpositive integer $m$, we exhibit an effectively computable bound for thesolutions $(x,y,\epsilon)$ of the diophantine equation $|F_\epsilon(x,y)|\leqm$.

Solving simultaneously Thue equations in the almost totally imaginary case