Davenport's theorem general case (Algebra by S Lang).

Asked Mar 12 '23 at 07:35

Active Mar 12 '23 at 08:21

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This is in regards to question 13 from chapter 4 of Lang. I got the case when $f,g$ are coprime, and for the general case I got the point where if $A$ is the gcd of $f$ and $g$, and $f'$, $g'$ are such, that, $Af'=f$, and $Ag'=g$, deg $f' \leq $ 2deg $((f')^3-(g')^2) - 2 \implies $ deg $f$ + 3 deg $A \leq $ 2deg $( f^2f' - g^2) - 2$. But I cannot show that deg$(f^2f'-g^2) \leq$ deg$(f^3-g^2)+ 3$deg$A$. Is the way I'm going about it wrong, I cannot see how to apply the hint Lang gave either.

Edit for Clarification The question being asked is to prove that deg$(f) \leq$ deg$(f^3 - g^2) - 2$, when $f,g$ are polynomials over an algebraically closed field. Question

edited Mar 12 '23 at 08:21

asked Mar 12 '23 at 07:35

David Pecherskiy

Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Mar 12 '23 at 07:41
Davenport's theorem follows immediately from Mason's $abc$ theorem for polynomials - which has a very short elementary proof. – Bill Dubuque Mar 12 '23 at 08:54

Davenport's theorem general case (Algebra by S Lang).

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