On 2015-01-26 14:22, Bruno Grenet wrote:
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].
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.
The second definition might be useful to have as algorithm, but I would
never call it xgcd(). Call it pseudo_xgcd() or resultant_xgcd().
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