Re: Thue equations

[John Cremona <john.cremona@gmail.com>]

On 14 November 2014 20:03, Karim Belabas
<Karim.Belabas@math.u-bordeaux.fr> wrote:
> * John Cremona [2014-11-14 20:34]:
>> Thanks for the replies, especially from Guillaume!  (G, I remember you
>> giving talks about your work on this many years ago, possibly when you
>> were still a student).
>>
>> Charles misunderstood the question slightly, I was wondering whether
>> the method used was the one which uses S-units (and this might have
>> depended on the degree; currently I am only interested in solving
>> degree 3 ones).
>
> Ariel Pacetti recently made the interesting remark that replacing the
> (arbitrary) model input by the user by a "reduced" one could have a
> dramatic effect. E.g.
>
>   ? thue(-1744357*x^3 - 9313887*x^2 - 16576896*x - 9834496, 1)
>   time = 8,264 ms.
>   %1 = []
>
>   \\ reduced form associated to the same cubic order
>   ? thue(196*x^3 - 468*x^2 - 207*x + 228, 1)
>   time = 32 ms.
>   %2 = []
>
> (Ariel's original example)
>

Ariel and I are working on a joint project -- this reduction of cubics
with negative discriminant is done using a gp script I wrote last week
implementing "juila reduction" -- which we all know is better than
"Matthewws reduction" in this case ;)


> I don't think this would affect your computations (I'd expect you
> to use reduced models, esp. in the cubic case :-), but I didn't have
> time to incorporate that trick yet.

I'm not sure what you thought "my computations" are then!

John

>
> Cheers,
>
>     K.B.
> --
> Karim Belabas, IMB (UMR 5251)  Tel: (+33) (0)5 40 00 26 17
> Universite de Bordeaux         Fax: (+33) (0)5 40 00 69 50
> 351, cours de la Liberation    http://www.math.u-bordeaux1.fr/~kbelabas/
> F-33405 Talence (France)       http://pari.math.u-bordeaux1.fr/  [PARI/GP]
> `
>