#8344: Factor constant polynomials over QQbar
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Reporter: boothby | Owner: AlexGhitza
Type: defect | Status: needs_info
Priority: critical | Milestone: sage-4.3.4
Component: basic arithmetic | Keywords:
Author: | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Comment(by dimpase):
Replying to [comment:8 cremona]:
> Good point, Paul; I had forgotten that no-one had done this yet, which
is silly since it only takes one line:
> {{{
> sage: x = polygen(QQbar)
> sage: f = 3*x^2-2
> sage: Factorization([(x-r,e) for r,e in
f.roots()],unit=f.leading_coefficient())
> (3) * (x - 0.8164965809277260?) * (x + 0.8164965809277260?)
> }}}
> But doing this for QQbar would also mean doing it for AAbar, which is a
little more complicated, which is probably why no-one has done it yet.
> {{{
> sage: f = x^4-2
> sage: x = polygen(AA)
> sage: f = x^4-2
> sage: fr = f.roots(QQbar)
> sage: f1 = Factorization([(x-r,e) for r,e in f.roots() if
f.imag().is_zero()],unit=f.leading_coefficient())
> sage: f2 = Factorization([(x^2-(r+r.conjugate()).real()*x+r.norm(),e)
for r,e in fr if f.image()>0])
> sage: f1*f2
> (x - 1.189207115002722?) * (x + 1.189207115002722?) * (x^2 +
1.414213562373095?)
> }}}
>
> It is very tempting to add this as a second patch to the ticket. Can
you see anything wrong in my worked examples here?
Shouldn't these roots be exact? (i.e. they should come with intervals
isolating them, otherwise one can potentially get
wrong results, mixing numerical noise with true multiplicities --- or is
it built-in f.roots() already?)
(same applies to QQbar---although I do not know anything
about isolating complex roots, excuse my ignorance...)
>
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8344#comment:11>
Sage <http://www.sagemath.org>
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