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babajijwerz 2022-05-30 Answered
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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Answers (1)

Gloletheods6g
Answered 2022-05-31 Author has 6 answers
I'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 get
1. | 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 .
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