#4000: [with patch, needs work] Implement QQ['x'] via Flint ZZ['x'] +
denominator
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Reporter: malb | Owner: somebody
Type: enhancement | Status: new
Priority: major | Milestone: sage-wishlist
Component: basic arithmetic | Keywords:
Reviewer: | Author: Martin Albrecht
Merged: |
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Comment(by spancratz):
Hi Martin,
I just started to look at the gcd methods again and I also looked at the
logic in polynomial_template.pxi. Here's the question:
Since the gcd of two polynomials is only defined up to multiplication by
rationals, what's the *right* way of dealing with this? I think one can
make a good argument for always returning the same normalisation. This
would also mean that we do *not* necessarily have gcd(a,0) == a. This is
currently the way it's handled in the file polynomial_template.pxi. If we
want to normalise the gcd, in which way should this be done? If it's non-
zero..
- Monic rational polynomial
- Primitive integer polynomial with positive leading coefficient
Of course, there are lots more but I think these two might be the most
sensible choices.
The first one has the advantage that it generalises to all polynomial
rings (over commutative rings with 1, at least). Upon adding a method
returning the monic scalar multiple of a polynomial to the template file,
one can still handle the cases of gcd(a,0) and gcd(0,b) in the template
file.
Personally though, I am more in favour of the second option, since this
might lead to faster code when working with QQ[]. In this case, we should
remove the handling of the above two cases from the template file and
always pass the call on to celement_gcd. This would mean that we leave
the normalisation up to the actual implementation of the polynomial ring,
rather than enforcing it across all base rings using the template file.
We would then also have to make sure that all celement_gcd methods are
happy to deal with zero arguments.
What do you think?
Sebastian
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/4000#comment:17>
Sage <http://sagemath.org/>
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