Karim Belabas on Thu, 13 Dec 2012 08:07:03 +0100

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Re: p-adic numbers

* 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?


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...


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/
F-33405 Talence (France)       http://pari.math.u-bordeaux1.fr/  [PARI/GP]