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

[Karim Belabas <Karim.Belabas@math.u-bordeaux.fr>]

* Aurel Page [2024-03-04 09:14]:
> 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).

Mentionig qflll() is important for a conceptual explanation: an
LLL-reduced "almost orthogonal" basis limits the number of linear
combinations to be considered.

But of course qfminim always incorporates the qflll step. You can apply
it directly to ellheightmatrix(E, [P,Q]) without bothering about reduction.

Cheers,

    K.B.
-- 
Pr. Karim Belabas, U. Bordeaux, Vice-président en charge du Numérique
Institut de Mathématiques de Bordeaux UMR 5251 - (+33) 05 40 00 29 77
http://www.math.u-bordeaux.fr/~kbelabas/