#15389: An algorithm for enumerating elements of bounded height in number fields
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Reporter: dkrumm | Owner: dkrumm
Type: enhancement | Status: needs_work
Priority: minor | Milestone: sage-6.3
Component: number theory | Resolution:
Keywords: sage-days55 | Merged in:
Authors: David Krumm, John | Reviewers: Ben Hutz
Doyle | Work issues:
Report Upstream: N/A | Commit:
Branch: | bac2d779a1cba38ec787e1a9cbfe77e609ccd99a
u/jdoyle/bdd_height | Stopgaps:
Dependencies: |
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Comment (by dkrumm):
Replying to [comment:12 bhutz]:
> - if bound is non-negative should always return 0, not empty list
(current skips 0 if the bound is < 1).
> - missed 0 for heights < 1
Maybe you're thinking of logarithmic height? For our height function, 0
has height 1.
> - if you're allowing real numbers for the matrix shouldn't you have the
base_ring for the polyhedron be RR?
Unfortunately, RDF is often not precise enough for our computations. The
polytope we deal with will sometimes have vertices with very small
coordinates, so that RDF thinks they are 0, and then things go wrong.
Ideally, we would be able to create a polyhedron whose base ring is a real
field with any given precision, but as far as I know this is not allowed
by the Polyhedron constructor. This is why we first compute the vertices
of our polytope with high precision as floating point numbers and then
convert them to rational numbers for the polyhedron computation. This is
not ideal, but otherwise there would be no point in allowing the user to
input a precision, since it's going to be cut down to 53 anyway.
> - does the precision test by changing to QQ ever actually fail. I'm not
sure why this is a precision test since any real number to some finite
number of decimals places can be converted to a rational.
It certainly does fail if the precision is not good enough. The issue is
that a fundamental unit can have an embedding with very very small
absolute value; when we take log of that, RR may interpret this as log(0).
If I recall correctly this is ok with RR, but when you try to coerce into
QQ it raises an error.
--
Ticket URL: <http://trac.sagemath.org/ticket/15389#comment:13>
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