Anyways, we should take into account that, even when both gcd and xgcd are
well defined, their definition coincide and everything is happy... there
are still defined only up to product of a unit. So there is yet another
possible source of inconsistency.
El domingo, 25 de enero de 2015, 21:05:00 (UTC+1), vdelecroix escribió:
>
> Hello,
>
> On sage-support somebody reported the following strange behavior
> sage: gcd(6/1,2/1)
> 2
> sage: xgcd(6/1,2/1)
> (1, 1/6, 0)
>
> I opened #17671 for that and it comes from the fact that there is a
> custom gcd for QuotientFields but no associated xgcd. I did it and it
> work fine.
>
> But, in order to get this behavior fixed once for all I introduced a
> test "_test_gcd_vs_xgcd" in the category CommutativeRings to check
> compatibility between gcd and xgcd if they are implemented. But it
> appear that some polynomial ring just fail:
>
> sage: TestSuite(Zmod(10)['a']).run()
> Failure in _test_gcd_vs_xgcd:
> ...
> AssertionError: The methods gcd and xgcd disagree:
> gcd(a,2*a^2 + 2) = 1
> xgcd(a,2*a^2 + 2) = (2, 8*a, 1)
> ------------------------------------------------------------
> The following tests failed: _test_gcd_vs_xgcd
>
> My question is: do we have a use case where gcd and xgcd should disagree?
>
> Vincent
>
> PS: On a related note, the following looks very wrong to me
> {{{
> sage: x = polygen(ZZ)
> sage: (x+2).gcd(x+4)
> 1
> }}}
>
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