#18836: Make refine_embedding into a method of number fields instead of
stand-alone
-------------------------------------+-------------------------------------
Reporter: cremona | Owner:
Type: enhancement | Status: needs_work
Priority: minor | Milestone: sage-6.8
Component: number fields | Resolution:
Keywords: | Merged in:
Authors: John Cremona | Reviewers: Vincent Delecroix
Report Upstream: N/A | Work issues:
Branch: | Commit:
u/cremona/18836-refine_embedding | ee94ab4b6da7a2bdf20b6fc50eefb2dba9190494
Dependencies: | Stopgaps:
-------------------------------------+-------------------------------------
Comment (by cremona):
This code was written while debugging the above. It runs fine unless you
comment out the two lines near the end. I would say that this is a bug in
QQbar.sqrt!
{{{
K.<i> = QuadraticField(-1)
# define a low-precision embedding from K to CC:
emb = K.embeddings(CC)[1]
# extend this to the closest embedding into QQbar:
old_gen = emb(K.gen())
rr = K.defining_polynomial().roots(QQbar, multiplicities=False)
diffs = [(CC(r)-old_gen).abs() for r in rr]
new_gen = rr[diffs.index(min(diffs))]
emb0 = K.hom([new_gen], check=False)
# Take a polynomial with 3 roots in K:
e1 = -4+i
e2 = 1+i
e3 = 3-2*i
print("Original ei: %s with parent %s" % ([e1,e2,e3],parent(e1)))
x = polygen(K)
pol = (x-e1)*(x-e2)*(x-e3)
# Find the roots again in QQbar:
pol0 = PolynomialRing(QQbar,'x')([emb0(c) for c in list(pol)])
e1, e2, e3 = pol0.roots(QQbar,multiplicities=False)
print("Roots ei: %s with parent %s" % ([e1,e2,e3],parent(e1)))
# Attempt to compute sqrt(e1-e2) from these:
d = e1-e2
print("d=%s with parent %s" % (d,d.parent()))
# If the next 2 lines are commented out, an error is raised in the sqrt!
s = d.imag().is_zero()
print("d.imag().is_zero()=%s" % s)
print("d=%s with parent %s" % (d,d.parent()))
d = d.sqrt()
print("d=%s" % d)
}}}
--
Ticket URL: <http://trac.sagemath.org/ticket/18836#comment:8>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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