Hello,
2015-01-26 14:22 UTC+01:00, Bruno Grenet <bruno.gre...@gmail.com>:
> In the special case of univariate polynomials over ZZ, I think there are
> two possibilities for xgcd:
>
> - either xgcd(p,q) = (g,u,v) where g = gcd(p,q) = up+vq with u,v in QQ[x];
> - or xgcd(p,q) = (g×r, u, v) where g = gcd(p,q), r = res(p,q), g×r =
> up+vq with u,v in ZZ[x].
Note that this second solution is precisely what we have in Sage right now
{{{
sage: x = polygen(ZZ)
sage: (x+2).xgcd(x+4)
(2, -1, 1)
}}}
Only the function `xgcd` is badly documented not the method of
polynomials. (And in the case when p and q are coprime over QQ, the
first term obtained from xgcd is the resultant.)
> More generally, Bézout coefficients seem to be usually defined only for
> PID, and this motivates the first solution (in general, replace ZZ by a
> UFD R and QQ by its field of fractions K). The second solution has the
> advantage of staying in the same polynomial ring, and it has a clear
> definition. I would go for the second solution, with an updated
> documentation.
Do you know in which context this second definition of Bézout
coefficients make sense? I guess we need the fraction field to be a
PID (or more generally a Bézout ring).
I am not tempted with a version of xgcd that has its member in another
ring... but for mysterious backward compatibility reasons we also have
{{{
sage: gcd(3.0, 6.0)
3
sage: gcd(3.0, 6.0).parent()
Integer Ring
}}}
And I would like the above behavior to be modified within #17671.
Vincent
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