On Sun, Nov 03, 2013 at 09:51:15AM -0800, Nils Bruin wrote:
> On Sunday, November 3, 2013 6:36:35 AM UTC-8, John Cremona wrote:
> >
> > The function is called global_height():
> >
> > sage: K.<a> = NumberField(x^3-2)
> > sage: b = a+1
> > sage: b.global_height()
> > 0.366204096222703
> >
> > John Cremona
> >
>
> That doesn't quite do the trick, though. For ABC, you'd need the height of
> the triple (a:b:c), not of the individual points. I think there are some
> tickets about heights for arithmetic dynamics? I'd think they would need
> naive height on projective space somewhere as well.
>
> It should be pretty straightforward to write something yourself:
> - clear denominators on your triple (a,b,c) [not necessarily minimally -
> if your ring of integers isn't a PID you might not be able to]
> - compute the archimedean contribution by taking max abs of the complex
> (and real) embeddings (take complex places twice)
> - divide by Norm (ideal generated by integral representatives)
>
Thank you for the replies.
Isn't it possible to define the quality only in
terms of the norm and the integer radical,
something like this:
q(a,b,c) = max( norm(a),norm(b),norm(c) ) /
(log(Delta(K)) + degree(K) * log(radical(norm(a*b*c))) )
with the restriction the norms of a,b,c to be coprime ?
Probably this needs some patching.
Is the ABC for this type of quality known and
does it make sense at all ?
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