Re: polred and variants

[Karim Belabas <Karim.Belabas@u-bordeaux.fr>]

* John Cremona [2026-03-17 16:38]:
> On Tue, 17 Mar 2026 at 15:34, John Cremona <john.cremona@gmail.com> wrote:
> >
> > Thanks again for this, Karim.   So (in the absolute case) the
> > canonical version is polredabs0() (*not) polredabs() which is
> > obsolete, while the non-canonical "best" version is polredbest().
> >
> > Out of interest, what is an example of a polynomial for which the
> > polredabs and polredbest reductions are different?
> 
> Ignore that question, I see that  x^4 + 2*x^2 - 12*x + 10 is unchanged
> by polredbest but becomes x^4+9 with polredabs.

Here's a "nicer" example from ??polredbest.

? P = X^12+8*X^8-50*X^6+16*X^4-3069*X^2+625;
? polredabs(P) == P
% = 1 \\ canonical

? poldisc(P)*1.
% = 1.2622 E55

? P = polredbest(P);
? poldisc(P)*1.      \\ ... but not best
% = 2.9012 E51

? P = polredbest(P); \\ ... far from it
? poldisc(P)*1.
% = 8.8704 E44

polredbest is not idempotent and may give better and better results.

Cheers,

    K.B.
-- 
Pr. Karim Belabas, U. Bordeaux, Vice-président en charge du Numérique
Institut de Mathématiques de Bordeaux UMR 5251 - (+33) 05 40 00 29 77
http://www.math.u-bordeaux.fr/~kbelabas/