Re: Computing p-adic logarithm with precision two

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Mon, Sep 09, 2024 at 04:24:52PM +0300, Georgi Guninski wrote:
> On Mon, Sep 9, 2024 at 4:13 PM Aurel Page <aurel.page@normalesup.org> wrote:
> >
> > Dear Georgi,
> >
> >
> > By "precision 2", do you mean computing the result up to O(p^2) or
> > O(p^3)? I assume that B is assumed to be in Z_p?
> > If the former, then p*(p-a) is unnecessary, -a*p is sufficient. If the
> > latter, then you are missing one term.
> >
> 
> Thanks.
> I mean with O(p^2) and my result is equal to pari's `log(B+O(p^2))`.
> 
> Does pari use the same algorithm?

PARI does:

  { /* compute log(x^(p-1)) / (p-1) */
    GEN q = gel(x,3), t = subiu(p, 1);
    a = Fp_pow(a, t, q);
    y = Fp_mul(Zp_log(a, p, e), diviiexact(subsi(1, q), t), q);
  }

which is similar when q=p^2.

Cheers,
Bill.