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Let   K   be an algebraic number field of degree    d  ⩾  3   ,σ  1   ,   σ  2   ,  …  ,   σ  dthe embeddings of   K   into   C  ,   α   a non‐zero element in   K  ,a  0   ∈  Z   ,a  0   >  0    andF   0    (  X  ,  Y  )  =   a   0∏    i  =  1    d    ⁡  (  X  −   σ   i    (  α  )  Y  )  .Let   υ   be a unit in   K  . For    a  ∈  Z   , we twist the binary formF  0    (  X  ,  Y  )   ∈  Z   [  X  ,  Y  ]by the powers    υ  a    (   a  ∈  Z   ) of   υ   by settingF   a    (  X  ,  Y  )  =   a   0∏    i  =  1    d    ⁡  (  X  −   σ   i    (  α   υ   a    )  Y  )  .Given    m  >  0   , our main result is an effective upper bound for the size of solutions(  x  ,  y  ,  a  )   ∈   Z  3of the Diophantine inequalities0  <   |    F   a    (  x  ,  y  )   |   ⩽  mfor which    x  y  ≠  0    and    Q  (  α   υ  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   υ  ; it also involves an effectively computable constant depending only on   d  .

FAMILIES OF THUE EQUATIONS ASSOCIATED WITH A RANK ONE SUBGROUP OF THE UNIT GROUP OF A NUMBER FIELD