Thanks for reporting the bug. I opened trac ticket #34733
<https://trac.sagemath.org/ticket/34733> for this.
On Tuesday, November 8, 2022 at 3:20:51 AM UTC-7 pkopr...@pkoprowski.eu
wrote:
> Hello everyone,
>
> I just discovered a bug in the polynomial division over quaternion
> algebras. I don't see it discussed anywhere on the internet, so I believe
> this is a proper place to report it.
>
> Bug description: the quo_rem method for polynomials with quaternionic
> coefficients outputs incorrect results. Moreover, there is no way to
> specify whether we want a right or left division. Here is a specific
> example.
>
> # Take the quaternion algebra (-1, -1)_QQ and the ring of polynomials over
> it:
> HH = QuaternionAlgebra(QQ, -1, -1)
> P.<x> = HH[]
> # Take two polynomials
> f = x^3 + HH.0*x + (HH.1 + 2*HH.2); print("f =", f)
> g = HH.2*x + HH.1 + 3; print("g =", g)
> # Try to compute the quotient and remainder
> q, r = f.quo_rem(g)
> # Check the correctness of the result assuming that this is RIGHT division
> print(q*g + r == f)
> # Check the correctness of the result assuming that this is LEFT division
> print(g*q + r == f)
>
> Both tests output False. Tested on Sage 9.5.
>
> Mathematical background: the division with the remainder for polynomials
> over division rings is well known and fully described e.g. in [1] even in a
> more general setting when the variable does not commute with the
> coefficients (in our case it does commute). So it is just a matter of
> proper implementation.
>
> *[1] Ore, Oystein*. Theory of non-commutative polynomials. *Ann. of Math.
> (2)* * 34 * (1933), no. 3, 480--508. MR1503119
> <https://mathscinet.ams.org/mathscinet-getitem?mr=1503119>
>
>
>
>
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