Hopefully this is better than what I asked yesterday and Milton solved.


Let $ C : F(x,y,z)=0$ be a projective genus $1$ curve over $\mathbb{Q}$ with
no restriction on the degree.


Write a point $P = (X , Y , Z)$ with the smallest coprime integers
$X,Y,Z$.


Is it true that for every fixed $ a > 1$


$$
\frac{\log \max(|X|,|Y|,|Z|)}{\log \min(|X|,|Y|,|Z|)} > a \qquad (1)
$$
finitely often?


Degrees $3,4$ are of interested too.




---


I believe Granville-Langevin conjecture (and $abc$) imply
some special cases.
$ x^3 + k y^3 = m z^3$ with coprime $x,y$ is just on the border of GL.


If (1) suitably fails for this curve GL fails for $ x y (x^3 + k y^3)$.



The relative sizes of coordinates of a point on projective genus 1 curve (second try)
Let $f$ be a nonconstant rational function on your curve $C$. For any point $P\in E(\mathbb{Q})$, write
$$
 f(P) = \frac{a\_f(P)}{b\_f(P)} \in \mathbb{Q}
$$
in lowest terms. Then Siegel's theorem implies that
$$
 \lim\_{P\in E(\mathbb{Q}), h(P)\to\infty} 
 \frac{\log|a\_f(P)|}{\log|b\_f(P)|} = 1.
$$
This looks pretty close to what you want, if you take (say) $f$ to be first $X/Y$, then $Y/X$, then $X/Z$, etc.


