Re: Question on completeness of the qfminin() command on finding all vectors for a given positive definite symmetric matrix

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

Dear Randall,

On 04/03/2024 23:31, American Citizen wrote:
> Carefull investigation of the elliptic curve points shows that M seems 
> to be missing some vectors.
>
> [ 0  0 -1 -1 -1 -1 -2 -2]           [ 1 0 2 ]
>                            MISSING:
> [-1 -2  0 -1  1  2  1  0]           [ 1 2 0 ]
>
> I obtained only 49 points, but there are actually 67 points of height 
> < 13 for curve E. I had to decompose these points as combinations of 
> the basis, to uncover what was missing here.
>
> Questions:
>
> 1. should I negate the columns of M and append them to M?  ( I am 
> guessing that the columns need to be unique) It is obvious that the 
> heights stay the same if we negate a column.
>
> 2. What about if M > 2 rows here, do the same, negate all columns of 
> n-rows for a basis of n-points?
>
> 3. Does qfminin exhaustively find all the vectors? or is just a 
> partial answer given as occurred in this case?
>
> This did take me 2-3 hours to troubleshoot.
Had you read the documentation, you would have known that qfminim 
returns the solutions up to negation.

Cheers,
Aurel