Ariel Pacetti on Fri, 14 Nov 2014 23:56:11 +0100


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Re: Thue equations
Actually Karim's example (which I sent him) has positive discriminant, and the reduction it uses is the standard sl_2 one for definite quadratic forms over the integers. For the indefinite ones (cubic fields with negative discriminants), John is right that Julia's reduction is faster than Matthew's and than reducing the binary indefinite form (for thue purposes) computationaly speaking (I tested billons of cases) 


Ariel

On Fri, 14 Nov 2014, John Cremona wrote:


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]
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