#14239: symbolic radical expression for algebraic number
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Reporter: gagern | Owner: davidloeffler
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-6.4
Component: number fields | Resolution:
Keywords: | Merged in:
Authors: Martin von Gagern | Reviewers: Marc Mezzarobba,
Report Upstream: N/A | Jeroen Demeyer
Branch: | Work issues:
u/gagern/ticket/14239 | Commit:
Dependencies: | 8ff313d2ea936471a4dda3989f9da440e8afe6b2
| Stopgaps:
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Comment (by jdemeyer):
Replying to [comment:46 gagern]:
> It seems as if in comment:42 you're suggesting that we should always
convert to `AA` resp. `QQbar` and do the equality comparison there.
No, it was a very general comment of the form "don't work around bugs". If
you say "I implemented this by doing B instead of the more obvious A,
since A is broken", then that's a sign that you should fix A first and
then use A.
> deciding whether to algebraic numbers are equal is hard
Really, why? Isn't it just a matter of comparing the minimal polynomials
first and then a floating-point interval approximation second? What's hard
about that?
> If I understand you correctly, in comment:43 you suggest that I wrap the
whole candidate collection process in a big loop which increments
precision until there is exactly one candidate left.
Exactly.
> The interval of the current algebraic number is known to be small enough
to be isolating. But by sheer bad luck it might still be big enough that
it almost touches a neighboring root, in which case increasing its
precision once might lower the need for precision when evaluating the
symbolic expressions. Suppose I got that wrapped up, I should hope that
I'd only have to increase precision a finite number of times to reduce the
number of candidates to one.
Well yes. I think a large enough precision should always work.
> But even then, I still haven't ruled out the case where I don't get
symbolic expressions for all roots.
Can that actually happen? I wouldn't mind to just give up in this case and
don't return a radical expression.
> In comment:44 I still disagree. When you convert number field elements,
you do that by plugging the converted generator into some polynomial.
You don't have to do that. You could just work with the minimal polynomial
of the number field element (which is trivial to compute).
--
Ticket URL: <http://trac.sagemath.org/ticket/14239#comment:49>
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