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) Cheers, Bill.