I agree that this needs to be fixed. Offhand, I don't know what the answer 
should be (for example, (1/(x*y), 0) also seems reasonable), but (0,1) is 
certainly not correct. So please do open a ticket. 

Please correct the typo when you make the ticket, though: q*y + r == x 
should be q*y + r == 1/x (in both places). 

On Sunday, January 17, 2021 at 8:25:27 AM UTC-7 Salvatore Stella wrote:

> Dear all,
> can someone please confirm that the current behaviour of sage is not the 
> expected one before I open a ticket about it?
>
> sage: R.<x,y> = LaurentPolynomialRing(QQ)
> sage: q,r = (1/x).quo_rem(y) ; q,r
> (0, 1)
> sage: q*y + r == x
> False
>
> The correct answer should have been:
>
> sage: R.<x,y> = LaurentPolynomialRing(QQ)
> sage: q,r = (1/x).quo_rem(y) ; q,r
> (0, 1/x)
> sage: q*y + r == x
> True
>
> Thanks
> S.
>

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