Re: polred and variants

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

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.

John

>
> 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/