We have F ( x , y ) ∈ Z [ X , Y ] a positive definite binary form of degree ≥ 3. I have to prove, without using lower bounds on linear forms in logarithms (we were working with Baker's theorems), that for each positive integer m the equation F ( x , y ) = m has only finitely many solutions in x , y ∈ Z . I also have to describe a method to find these.We know that the coefficient of X d of F is positive, and that all zeros of F ( X , 1 ) are in C ∖ R . We also know the discriminant of F is negative.I have trouble finding this out by myself. I also looked up some number theory books, but most literature is about binary quadratic forms, not the binary form I have to prove this for. Hope someone can help me!

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We have   F  (  x  ,  y  )  ∈      Z    [  X  ,  Y  ] a positive definite binary form of degree   ≥  3. I have to prove, without using lower bounds on linear forms in logarithms (we were working with Baker&#039;s theorems), that for each positive integer m the equation   F  (  x  ,  y  )  =  m has only finitely many solutions in   x  ,  y  ∈      Z  . I also have to describe a method to find these.We know that the coefficient of       X    d   of   F is positive, and that all zeros of   F  (  X  ,  1  ) are in       C    ∖      R  . We also know the discriminant of   F is negative.I have trouble finding this out by myself. I also looked up some number theory books, but most literature is about binary quadratic forms, not the binary form I have to prove this for. Hope someone can help me!

Gloletheods6g · Accepted Answer

I&#039;d say you mean a degree   d homogeneous polynomial   f  ∈      Z    [  X  ,  Y  ] having no real zeros other than (0,0).With   c  (      X    d    )  ,  c  (      Y    d    ) the coefficients of       X    d    ,      Y    d   and  A  =  min  (      |    c  (      X    d    )      |    ,      |    c  (      Y    d    )      |    ,      inf          t      ∈              R                  |    f  (  1  ,  t  )      |    ,      inf          t      ∈              R                  |    f  (  t  ,  1  )      |    )you get1.       |    f  (  x  ,  0  )      |    ≥  A      |    x            |        d    ,      |    f  (  0  ,  y  )      |    ≥  A      |    y            |        d  2. if   y  ≠  0,       |    f  (  x  ,  y  )      |    =      |    y            |        d        |    f  (  x      /    y  ,  1  )      |    ≥  A      |    y            |        d  ,3. if   x  ≠  0,       |    f  (  x  ,  y  )      |    =      |    x            |        d    f  (  1  ,  y      /    x  )      |    ≥  A      |    x            |        d   ie.      |    f  (  x  ,  y  )      |    ≥  max  (  A      |    x            |        d    ,  A      |    y            |        d    )which implies that   f  (  X  ,  Y  )  −  m has finitely many zeros   ∈            Z        2  .

We have F ( x , y ) ∈ <mrow class="MJX-TeXAtom-ORD">

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