Re: nfgaloisconj

[macsyma <macsyma@yahoo.co.jp>]

> Bill Allombert on Wed, 31 Jul 2019 11:06:53 +0200

Thank you, Bill.

> I was expecting that if the degree of g get large, finding 
> p would be costly.

I think too that your method is superior 
if we consider the root separation of a polynomial having higher degree.
However, in consideration of the range of degree that can be handled in practice (<=10^3?) , 
it might not be able to demonstrate its ability.


The following is a timing data for f = x^n-2.

 [n,d,processing time of [G12p(f), nfsplittingaut(f)]]
=[11,110,[465,710]],[12,48,[147,161]],[13,156,[1018,1600]],[14,84,[517,575]],[15,120,[828,1069]],
 [16,64,[325,367]],[17,272,[6317,11165]],[18,108,[1544,1716]],[19,342,[11107,24051]],[20,160,[4102,4457]],
 [21,252,[7077,9658]],[22,220,[8203,9252]],[23,506,[33323,89064]],[24,96,[788,866]],[25,500,[38203,78668]],
 [26,312,[24192,27487]],[27,486,[43061,81301]],[28,336,[33914,38605]],[29,812,[144222,388746]],[30,240,[9641,11226]],
 [31,930,[255020,626746]],[32,256,[19722,23111]],[33,660,[89254,193020]],[34,544,[151791,180759]],[35,840,[217197,409017]],

> Bill Allombert on Wed, 31 Jul 2019 14:01:56 +0200

> 2) if the order of the Galois group is not given and the group is not
> the symmetric group, then nfsplitting already computes a large part of
> nfisincl to check that the tower stops, so we could arrange for
> nfsplitting to return also the embeddings.That is good news.

That is good news.

macsyma