Re: another simple question

Given [x,y], the set of linear linear combinations between [y^2,xy,y,x^3,x^3,x,1] is a vector space of dimension 6.

The integer solutions form a lattice.

It is not difficult to see that there exist some solutions that are bounded by |x| and |y|.

I would not be surprized if someone could prove the existence of a solution not larger than |x|^(1/2) or maybe |x|^(1/6).

The true question is whether there are solutions that are much smaller than these generic bounds,

whether the lattice contains a very small nonzero point.

LLL is a good algorithm to find a short vector in a lattice. This is why the function lindep uses it !

Randall, in your question, it is difficult to be much smaller than 1/2 or 5/8.

However, in the example of Bill, the solution is highly smaller than |x| and |y|.

In some sense, the solution becomes unique !

Denis.

De: "John Cremona" <john.cremona@gmail.com>
À: "American Citizen" <website.reader3@gmail.com>
Cc: "pari-users" <pari-users@pari.math.u-bordeaux.fr>
Envoyé: Mercredi 17 Avril 2024 11:30:48
Objet: Re: another simple question

Is it really a sensible question to ask? You can assign arbitrary values to all coefficients except the constant term and then solve for that. For example, set k=y^-x^3, then (x,y) is on y^2=x^3+k.
I know that this is not exactly in the spirit of the question.

John

On Wed, 17 Apr 2024 at 04:45, American Citizen <website.reader3@gmail.com> wrote:
Recently, we saw a point with thousands of digits posted with the
question, what elliptic curve might have this point?

The answer was to use lindep to determine the elliptic curve, and so
everything went fine.

However, I am dubious that this general approach using lindep on [y^2,
xy, y, x^3, x^2, x, 1] will work when we have points of lesser heights.

Suppose we have the point [1/2, 5/8] ??

I strongly suspect that thousands perhaps millions, maybe more? elliptic
curves can be found containing this point.

How can we effectively pare down the possibilities to something we can
work with?

Randall