Bill Allombert on Sat, 15 Sep 2012 11:48:09 +0200

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Re: relative saturation?

On Fri, Sep 14, 2012 at 06:46:46PM -0500, Daniel Allcock wrote:
> Dear all,
> I have a basis for a saturated subgroup L of Z^n, and one extra vector V in
> Z^n but not in L.  I'd like to find an element of Z^n which together with L
> generates the saturation of the span of L and V.  I think of this as the
> "saturation of <L,V> relative to L".
> I can't find any ready-made function that lends itself to an easy
> implementation of this.  The best I can think of right now is to find a basis
> for the saturation of <L,V>, express it in terms of V and my basis for L,
> look at the V-components, and follow Euclid's algorithm to find a Z-linear
> comb whose V-component is smallest possible.  This looks icky and I am
> probably missing something more obvious.  

I would suggest you look first at the GP function matrixqz().
I would start by expressing V as an element W of L\otimes Q and use 
matrixqz(,-2) to compute the saturation of [W|Z^n].