Pankaj Charpe charpe.pankajamol at gmail.com
Mon Feb 5 07:19:04 CET 2018

```I am clear with this now. Thanks for the explanation.

-Pankaj charpe

On Mon, Feb 5, 2018 at 1:28 AM, Emmanuel ThomÃ© <Emmanuel.Thome at inria.fr>
wrote:

> On Sat, Feb 03, 2018 at 08:12:49PM +0530, Pankaj Charpe wrote:
> > Thanks for the reply. I have studied your thesis. p and r (roots) are
> clear
> > to me but I am not getting any explanation for those n1 and n2. Can you
> > elaborate their use? I would really appreciate your reply. I also mailed
> > via mailing list.
> >
> > Thanks & Regards
> > Pankaj Charpe
>
> Hi,
>
> Let's say we have:
>
> 5: 1,3
> 25:2,1: 6,13
>
> All pairs with a-1*b = 0 mod 5 or a-3*b = 0 mod 5 must receive a
> contribution equal to round(log(5)).
>
> Pairs with a-6*b = 0 mod 25 (which implies, in particular, a-1*b = 0 mod
> 5) receive an extra contribution of round(2*log(5))-round(1*log(5)).
>
> A polynomial with this behaviour could be, for example (two distinct
> examples below):
>     sage: ((x-6)*(x-13)+25).expand()
>     x^2 - 19*x + 103
>     sage: ((x-6)*(x-13)*ZZ['x'](GF(5)['x'].irreducible_element(2))+
> 25).expand()
>     x^4 - 15*x^3 + 4*x^2 + 274*x + 181
>
> And it can go on and on, as you lift to higher 5-adic roots. Unramified
> roots in the p-adics will follow a simple pattern of this kind.
>
> Now there are cases for which we need more information. Consider for
> example the polynomial:
>     sage: ((x-6)*(x-1)+25).expand()
>     x^2 - 7*x + 31
>
> For a-1*b = 0 mod 5, the 5-valuation goes from 0 to 2. We write this as:
>
>     5:2,0: 1
>
> while for higher powers we would write:
>
>     25:3,2: 1,6
>
> All sorts of situations are possible. The factor base format is meant to
> express the different things that can occur in down-to-earth terms.
>
> E.
>
>
>
>
>
>
>
> > On Feb 3, 2018 5:30 PM, "Emmanuel ThomÃ©" <emmanuel.thome at inria.fr>
> wrote:
> >
> > > Yes.
> > > Actually log(p) would be less misleading than degree(p)...
> > > E.
> > >
> > >
> > > On February 3, 2018 11:21:24 AM GMT+01:00, Pierrick Gaudry <
> > > pierrick.gaudry at loria.fr> wrote:
> > > >Hi,
> > > >
> > > >From an old README file I have somewhere in an old directory:
> > > >
> > > >    Factor base file format:
> > > >    ------------------------
> > > >
> > > >    An entry is of the form:
> > > >
> > > >    q:n1,n2: r1,r2,r3
> > > >
> > > >    In the (frequent) case where n1,n2=1,0 this can be abridged with:
> > > >
> > > >    q: r1,r2,r3
> > > >
> > > >Here, q is a irreducible or a irreducible power, ri are the
> > > >corresponding
> > > >roots and the contribution that must be subtracted at these positions
> > > >is
> > > >    (n1-n2)*degree(p) (assuming smaller powers of this irreducible
> have
> > > >  alredy been taken care of).  By position, we mean (a,b) such that a
> -
> > > >    b*ri = 0 mod q.
> > > >
> > > > The roots ri must be sorted in lexicographical order.  If a root ri
> is
> > > >    greater or equal to q, it means that this is a projective root:
> > > >    subtracting q gives a root for the reciprocal polynomial (or
> > > >    equivalently, (1:(ri-q)) is the projective root).
> > > >
> > > > It is allowed to have several lines with the same q, but there must
> be
> > > >    only one line for a given (q,n1,n2) triple.
> > > >
> > > >Hopefully this is still valid in the version you are using.
> > > >
> > > >Regards,
> > > >Pierrick
> > > >
> > > >On Sat, Feb 03, 2018 at 01:50:46PM +0530, Pankaj Charpe wrote:
> > > >> Hi,
> > > >>  In factor base construction of cado-nfs we have this entry,
> > > >>                             Factor Base format: q:n1,n2:r1,r2,r3
> > > >>
> > > >> Can you please explain me what is these n1 and n2 ?. I will be very
> > > >> thankful to you.
> > > >>
> > > >>
> > > >> Thanks & Regards
> > > >> Pankaj charpe
> > > >
> > > >> _______________________________________________
> > > >> Cado-nfs-discuss mailing list
> > > >> Cado-nfs-discuss at lists.gforge.inria.fr
> > > >
> > > >_______________________________________________
> > > >Cado-nfs-discuss mailing list
> > > >Cado-nfs-discuss at lists.gforge.inria.fr