Re: trying to parameterize solutions for Pythagorean ratios and Diophantine m-tuples

[Aurel Page <aurel.page@normalesup.org>]

Dear Randall,

On 04/03/2024 02:18, American Citizen wrote:
> They state (for a rank 2 curve) with Mordell-Weil basis P and Q
>
> that all rational points are a composition of
>
> { uP + vQ for u,v in Z }
>
> Does this mean that some weird combination of 1000000 * P + 938471*Q 
> might produce a point of low height?
It depends on the choice of the initial P and Q. The canonical height is 
positive definite quadratic form that gives the Mordell-Weil group mod 
torsion the structure of a Euclidean lattice. If you take {P,Q} to be a 
reduced basis, then there cannot be a large linear combination that 
magically produces a point of low height. However, that could happen if 
{P,Q} is a very bad basis. If you want all points of bounded height, you 
should first compute a reduced basis (with qflll) and then use qfminim 
to enumerate all points of bounded height (don't forget to take 
negatives and add torsion points at the end).

Cheers,
Aurel