  Twisting a binary form $F_0(X,Y)\in{\mathbb{Z}}[X,Y]$ of degree $d\ge 3$ by
powers $\upsilon^a$ ($a\in{\mathbb{Z}}$) of an algebraic unit $\upsilon$ gives
rise to a binary form $F_a(X,Y)\in{\mathbb{Z}}[X,Y]$. More precisely, when $K$
is a number field of degree $d$, $\sigma_1,\sigma_2,\dots,\sigma_d$ the
embeddings of $K$ into $\mathbb{C}$, $\alpha$ a nonzero element in $K$,
$a_0\in{\mathbb{Z}}$, $a_0>0$ and $$ F_0(X,Y)=a_0\displaystyle\prod_{i=1}^d
(X-\sigma_i(\alpha) Y), $$ then for $a\in{\mathbb{Z}}$ we set $$
F_a(X,Y)=\displaystyle a_0\prod_{i=1}^d (X-\sigma_i(\alpha\upsilon^a) Y). $$
Given $m\ge 0$, our main result is an effective upper bound for the solutions
$(x,y,a)\in{\mathbb{Z}}^3$ of the Diophantine inequalities $$ 0<|F_a(x,y)|\le m
$$ for which $xy\not=0$ and ${\mathbb{Q}}(\alpha \upsilon^a)=K$. Our estimate
involves an effectively computable constant depending only on $d$; it is
explicit in terms of $m$, in terms of the heights of $F_0$ and of $\upsilon$,
and in terms of the regulator of the number field $K$.
