Thue and unit equations over totally real number fields

We obtain a polynomial type upper bound for the size of the integral solutions of Thue equations $F(X,Y) = m$ defined over a totally real number field $K$, assuming that $F(X,1)$ has at least a non real root and, for every couple of non real conjugate roots $(\alpha, \bar{\alpha})$ of $F(X,1)$, the field $K(\alpha, \bar{\alpha})$ is a CM-field. In case where $F(X,1)$ has also real roots, our approach gives polynomial type bounds that the Baker's method was not able to provide other than exponential bounds. Using this result, we derive an improved upper bound for the size of the solutions of the unit equation defined over a totally real number field, which allows us to deduce an upper bound for the size of the integral solutions of Thue equations defined over a totally real number field.