On Monday, November 4, 2013 1:29:10 AM UTC-8, Georgi Guninski wrote: > > 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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