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Let  K  be a field of characteristic    p  &gt;  0    and let  G  be a subgroup ofK    *    ×    K    *withdim    ℚ    (  G    ⊗    Z    ℚ  )  =  r    finite. Then Voloch proved that the equation    a  x  +  b  y  =  1  in  (  x  ,  y  )  ∈  G    for given    a  ,  b  ∈    K   ⁎has at mostp   r   (    p   r   +  p  −  2  )  /  (  p  −  1  )    solutions    (  x  ,  y  )  ∈  G   , unless(  a  ,  b  )   n   ∈  G    for some    n  ≥  1   . Voloch also conjectured that this upper bound can be replaced by one depending only on  r . Our main theorem answers this conjecture positively. We prove that there are at most    31    ×19    r  +  1solutions ( x ,  y ) unless(  a  ,  b  )   n   ∈  G    for some    n  ≥  1    with ( n ,  p )  = 1. During the proof of our main theorem, we generalize the work of Beukers and Schlickewei to positive characteristic, which heavily relies on diophantine approximation methods. This is a surprising feat on its own, since usually these methods cannot be transferred to positive characteristic.

On the equation X1 + X2 = 1 infinitely generated multiplicative groups in positive characteristic