Pascal Molin on Wed, 24 Oct 2012 14:01:22 +0200

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Re: charpoly via Newton

> However flag 1 and 2 are even faster.
indeed ! I agree with Karim that charpoly deserves better input detection.

>> Dans tous les cas, je souhaiterais proposer pour Pari un certain
>> nombre d'opÃrations sur les polynÃmes (reconstruction à partir
>> des sommes de Newton, produits composÃs, puissances symÃtriques --
>> c'est initialement Ãa que je voulais --...), modalitÃs et
>> syntaxe à voir par exemple durant l'atelier de janvier.
> A lot of them arealready available. Maybe it is missing a GP interface.
> At least:
> polsym(x,n): column vector of symmetric powers of the roots of x up to n.

I think translation from and to newton sums should be available in gp with
easy names (personnaly I do not like polsym), and in any characteristic.
This is very useful.

pol2newton and newton2pol, for example, or poltonewton and polfromnewton
(I think it is nice that all function primarily related to pols start with pol)

The others I mention are

polcomposed(P,Q) = prod_{P(a)=Q(b)=0} (x-ab)

and if P(x) = prod_{i<=d} (x-xi)

polsympow(P,r) = prod_{i1<=i2<=... ir<=d} (x- xi1xi2...xir)

which appears in symmetric powers of representations.
I have formulas based on Newton sums for the latter.