Karim Belabas on Thu, 13 Dec 2012 08:54:57 +0100 |
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Re: p-adic numbers |
* 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/ F-33405 Talence (France) http://pari.math.u-bordeaux1.fr/ [PARI/GP] `