Bill Allombert on Mon, 01 Nov 2010 18:31:15 +0100

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Re: Q re polcyclo() etc

On Tue, Oct 26, 2010 at 09:05:31PM +0200, Bill Allombert wrote:
> 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.

Kurt pointed out that this inegality follow from the arithmetic-geometric mean inequality,
and that equality can only occurs for roots of unities.

Some other results:
First, if K(alpha) is totally real or is abelian over Q, then T2(alpha) is an integer.
This apply in particular for polsubcyclo()/galoissubcyclo().

Also, there are formula for polsubcyclo(p,k) for k<=4 and p prime
for example 
polsubcyclo(p,2) = x^2+x+(1-kronecker(-1,p)*p)/4

If K is imaginary quadratic, then T2(alpha)=2*Norm(alpha) 
So if p=3 [mod 4] then
polredabs(polsubcyclo(p,2)) == x^2-x+(1+p)/4 == subst(polsubcyclo(p,2),x,-x)