#20086: rational powers in ZZ[X] and QQ[X]
-------------------------------------+-------------------------------------
Reporter: cheuberg | Owner:
Type: defect | Status: needs_info
Priority: major | Milestone: sage-7.2
Component: basic arithmetic | Resolution:
Keywords: | Merged in:
Authors: Clemens | Reviewers: Benjamin Hackl,
Heuberger, Vincent Delecroix, | Vincent Delecroix
Benjamin Hackl | Work issues:
Report Upstream: N/A | Commit:
Branch: public/20086 | 6f91df97a81fa094c04583314ad38cd5fd199cdb
Dependencies: | Stopgaps:
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Comment (by behackl):
Replying to [comment:58 nbruin]:
> Taking an n-th root of an element in a ring R by computing a
factorization is an insane way to go about it. A much better generic
strategy is to hope that the univariate polynomial ring over R has a root
finding algorithm and see if the polynomial `x^n-a` has a root. You will
see that:
> - it actually has a decent performance over QQ (although there the
algorithm should really be special-cases)
> - it will work over most fields, including the ones that are not
constructed as fraction fields of rings with a factorization algorithm.
> - you don't have to mess around with the unit part that a factorization
algorithm probably won't recognize.
While I like the general idea of this approach, I'm for discussing it on a
follow-up ticket; see it as a "performance enhancement" of this
implementation. Also, I'm not sure how exactly the root-finding algorithm
for these univariate polynomial rings can be properly motivated, for me
neither `roots()` nor `any_root()` did the job:
{{{
sage: R.<x> = QQ[]
sage: P.<X> = R[]
sage: a = X^2 - (x^2 + 2*x + 1)
sage: a.any_root(R)
Traceback (most recent call last):
...
TypeError: Unable to coerce Principal ideal (1) of Univariate Polynomial
Ring in x over Rational Field (<class
'sage.rings.polynomial.ideal.Ideal_1poly_field'>) to Rational
}}}
Do you strongly object against setting this back to `positive_review`?
--
Ticket URL: <http://trac.sagemath.org/ticket/20086#comment:59>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
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