Re: polred and variants

[John Cremona <john.cremona@gmail.com>]

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canonical version is polredabs0() (*not) polredabs() which is
obsolete, while the non-canonical "best" version is polredbest().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?

John

On Sat, 24 Jan 2026 at 14:07, Karim Belabas <Karim.Belabas@u-bordeaux.fr> wrote:
>
> * John Cremona [2026-01-22 22:19]:
> [...]
> > But it is still true that there is no documentation in this manual for
> > any non-obsolete *polred* functions.
>
> Indeed, here is the complete list of "polred" public functions
> (from "grep polred parideclh.h") with comments
>
> // OBSOLETE, don't use them in new programs
> GEN     factoredpolred(GEN x, GEN fa);
> GEN     factoredpolred2(GEN x, GEN fa);
> GEN     polredord(GEN x);
> GEN     polred(GEN x);
> GEN     polred0(GEN x, long flag);
> GEN     polred2(GEN x);
> GEN     polredabs(GEN x);
> GEN     polredabs2(GEN x);
> GEN     polredabsall(GEN x, long flun);
> GEN     smallpolred(GEN x);
> GEN     smallpolred2(GEN x);
> GEN     rnfpolred(GEN nf, GEN pol, long prec);
>
> // Expensive but provide canonical representatives
> GEN     polredabs0(GEN x, long flag);
> GEN     rnfpolredabs(GEN nf, GEN pol, long flag);
>
> // Fast, generally best. Possibly smaller discriminant (!)
> GEN     polredbest(GEN x, long flag);
> GEN     rnfpolredbest(GEN nf, GEN R, long flag);
>
> The documentation for the 4 non-obsolete functions is found in ??polredabs,
> ??polredbest, ??rnfpolredabs, ??rnfpolredbest. Or gp user's manual.
>
> > One suggested improvement to the gp documentation for nf_ORIG: it
> > seems to me that the second thing returned by the nf_ORIG flag is
> > Mod(a,P) and not a itself: in the code I have to use lift() to get the
> > polynomial.
>
> Indeed. The litteral wording was correct (nothing was said about the
> type of 'a' and Mod(a,P) is 'a' if already a t_POLMOD mod P) but misleading.
>
> Now clarified in 'master'. :-)
>
> Thanks !
>
>     K.B.
> --
> Pr. Karim Belabas, U. Bordeaux
> Institut de Mathématiques de Bordeaux UMR 5251 - (+33) 05 40 00 29 77
> http://www.math.u-bordeaux.fr/~kbelabas/