Let
$K$
be an algebraic number field of degree
$d\geqslant 3$
,
$\unicode[STIX]{x1D70E}_{1},\unicode[STIX]{x1D70E}_{2},\ldots ,\unicode[STIX]{x1D70E}_{d}$
the embeddings of
$K$
into
$\mathbb{C}$
,
$\unicode[STIX]{x1D6FC}$
a non-zero element in
$K$
,
$a_{0}\in \mathbb{Z}$
,
$a_{0}>0$
and
$$\begin{eqnarray}F_{0}(X,Y)=a_{0}\mathop{\prod }_{i=1}^{d}(X-\unicode[STIX]{x1D70E}_{i}(\unicode[STIX]{x1D6FC})Y).\end{eqnarray}$$
Let
$\unicode[STIX]{x1D710}$
be a unit in
$K$
. For
$a\in \mathbb{Z}$
, we twist the binary form
$F_{0}(X,Y)\in \mathbb{Z}[X,Y]$
by the powers
$\unicode[STIX]{x1D710}^{a}$
(
$a\in \mathbb{Z}$
) of
$\unicode[STIX]{x1D710}$
by setting
$$\begin{eqnarray}F_{a}(X,Y)=a_{0}\mathop{\prod }_{i=1}^{d}(X-\unicode[STIX]{x1D70E}_{i}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D710}^{a})Y).\end{eqnarray}$$
Given
$m>0$
, our main result is an effective upper bound for the size of solutions
$(x,y,a)\in \mathbb{Z}^{3}$
of the Diophantine inequalities
$$\begin{eqnarray}0<|F_{a}(x,y)|\leqslant m\end{eqnarray}$$
for which
$xy\not =0$
and
$\mathbb{Q}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D710}^{a})=K$
. Our estimate is explicit in terms of its dependence on
$m$
, the regulator of
$K$
and the heights of
$F_{0}$
and of
$\unicode[STIX]{x1D710}$
; it also involves an effectively computable constant depending only on
$d$
.