#6046: [with patch, needs review] Implement local and global heights for number
field elements
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Reporter: cremona | Owner: was
Type: enhancement | Status: new
Priority: major | Milestone: sage-4.0.1
Component: number theory | Keywords: number field height
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Comment(by fwclarke):
Replying to [comment:2 cremona]:
> We must sort out this mess with the embeddings. Your first doctest
failure (with different behaviour in different runs) must mean that the
ordering of the embeddings is not deterministic. That must be fixed, say
by ordering the roots when the embeddings are found. Secondly we must
have a fool-proof way of determining which embeddings are real when you do
K.complex_embeddings(). What I would prefer is to have a function called
(perhaps) K.archimedean_completions() which returns a list of r+s
embeddings, the first r into a RealField and the last s into a
ComplexField (or even a list of two lists of embeddings, of lengths r and
s respectively). That way the codomain of a real embedding would be a
RealField which could easily be tested for. While we are at it the
complex list would only contain one of each pair.
>
> So, how to implement this? Not problem with the list of real
embeddings, though they should be sorted by the natural real ordering on
the image og K.gen(); for the non-real ones we could first find all
embeddings into CC, then sort them by their imaginary parts and then take
the last s of these (where s could be defined to be (n-r)/2 where n is the
degree and r the number of real embeddings).
>
> Does that sound workable?
Yes; no time to experiment more with this, but I note that `places` in
`number_field.py` seems to do what's needed:
{{{
def places(self, all_complex=False, prec=None):
"""
Return the collection of all places of self. By default, this
returns the set of real places as homomorphisms into RIF first,
followed by a choice of one of each pair of complex conjugate
homomorphisms into CIF.
On the other hand, if prec is not None, we simply return places
into RealField(prec) and ComplexField(prec) (or RDF, CDF if
prec=53).
...
}}}
so that
{{{
sage: K.<a> = NumberField(x^4+3*x^2-17)
sage: K.places()
[Ring morphism:
From: Number Field in a with defining polynomial x^4 + 3*x^2 - 17
To: Real Field with 106 bits of precision
Defn: a |--> -1.699259307373674805202512065354,
Ring morphism:
From: Number Field in a with defining polynomial x^4 + 3*x^2 - 17
To: Real Field with 106 bits of precision
Defn: a |--> 1.699259307373674805202512065354,
Ring morphism:
From: Number Field in a with defining polynomial x^4 + 3*x^2 - 17
To: Complex Field with 53 bits of precision
Defn: a |--> 2.42641344244876*I]
}}}
Not defined (yet!) for relative number fields.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/6046#comment:4>
Sage <http://sagemath.org/>
Sage - Open Source Mathematical Software: Building the Car Instead of
Reinventing the Wheel
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