Re: Trace problem

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

On Sat, Nov 10, 2012 at 08:15:50AM -0700, Kurt Foster wrote:
> Let K = bnfinit(T)) (T monic and irreducuble in Z[x] of degree n >
> 1, with Galois group cyclic of order n) be a number field, R = K.zk
> its ring of integers, Mod(x,T) a non-zero element of R.  I want to
> determine all units u such that trace(Mod(x*u,T)) is zero.
> 
> It's certainly possible to obtain a Z-basis of the set Y of Mod(y,T)
> in R for which
> 
> trace(Mod(x*y, T)) = 0.
> 
> But how to determine which units (if any) are in Y has me flummoxed.
> Perhaps I am overlooking something obvious.  I note that Y only has
> rank n-1, one less than the rank n of R.

I will give you an example (though non Galois).

Solve x^3-2^y^3=+/-1 with x,y in Z is equivalent to finding units in Z[a]/(a^3-2)
which lie in the hyperplane generate by 1 and a. The unit group of this ring is
of rank 1.  
While I know at least three different proofs that the only solutions are 
(+/-1,0) and +/-(1,1), none of them are straight forward:
1) an irrationality measure for 2^(1/3) (Roth theorem imply there are only finitely many
solution).
2) x^3-2^y^3=1 is Thue equation that can be dealt with thueinit
3) x^3-2^y^3=1 is an elliptic curve (36a3) of rank 0.

For n=3, your problem is equivalent to solving an equation of the 
form P(x,y)=+/-1 with P of degree 3. Then you can use one of the method
above.

More general equation involving units are hard to solve.
There are lot of papers by M. Pohst, A. Petho, I. Gaal, A. Berczes and other
on this topic.

Cheers,
Bill.