Re: p-adic numbers

[Karim Belabas <Karim.Belabas@math.u-bordeaux1.fr>]

* Karim Belabas [2012-12-13 08:07]:
> * somayeh didari [2012-12-12 19:34]:
> > I want to know, if there is a command for determining whether a
> > multivariate polynomial has a solution in p-adic numbers?
> 
> No. 
> 
> There's not even a command to determine whether a hypersurface has an
> Fp-rational point. If p and the number n of variables are small, you can
> write a naive program doing an exhaustive search for Fp-points using
> forvec (set n-1 variables, then use polrootsmod):
> 
> - if there's no Fp-point, there are no solutions
> 
> - if there's one smooth Fp-point, apply (multivariate) Hensel/Newton
> 
> - if all Fp-points are singular, tough luck...
[ I had to go before finishing. Here's the end ]

- if all Fp-points are singular, tough luck... For each such singular
point (a_1,...,a_n), change variables by setting x_i = a_i + p * y_i,
divide by content (at least p) and solve the new equation in (y_i) for a
p-adic point. Eventually, you will either find a smooth point [ which lifts ]
or rule out all singular points.

There are better approaches for specific classes of varieties. Please be
more specific if the above does not work for you. :-)

Cheers,

    K.B.
--
Karim Belabas, IMB (UMR 5251)  Tel: (+33) (0)5 40 00 26 17
Universite Bordeaux 1          Fax: (+33) (0)5 40 00 69 50
351, cours de la Liberation    http://www.math.u-bordeaux1.fr/~belabas/
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