Re: Q re polcyclo() etc

[Bill Allombert <Bill.Allombert@math.u-bordeaux1.fr>]

On Mon, Oct 18, 2010 at 03:01:35PM -0600, Kurt Foster wrote:
> I've noticed that often (but not always), if f is the output of
> polcyclo(), polsubcyclo(), or galoissubcyclo(), then f ==
> polredabs(f).  What is known about when this happens?

Let K a number field of degree n with complex embeddings (sigma_1,...,sigma_n).
Let T2(alpha)=sum_i |sigma_i(alpha)|^2. This is a positive quadratic form over Z_K.

polredabs(K) returns minpoly(alpha) for one alpha in O_K such that T2(alpha) is minimal and
minpoly(alpha) has degree n. Which alpha exactly is used is a matter of sorting the polynomials
lexicographically, etc.

The only theoretical result I know is that T2(alpha)>=n for all fields and all
integral alpha!=0 (Unfortunately I do not remember the proof).
This minimum is attained for roots of unity.

Given that cyclotomics polynomials are small lexicographically, 
we have polredabs(f)=f for polcyclo(2*p) and polcyclo(2^n).

For polsubcyclo(f,d) then T2(alpha)=eulerphi(f), so if eulerphi(f)/d is small, then it 
is likely that either alpha or -alphe is minimal for T2.

Cheers,
Bill.