datapoint: this still seems to work fine in 10.5.beta6 and indeed takes a fraction field for the scalar:

sage: K.<z> = PowerSeriesRing(QQ,default_prec=10)
sage: R.<x> = PowerSeriesRing(K,default_prec=6)
sage: x/(1-z)
(1 + z + z^2 + z^3 + z^4 + z^5 + z^6 + z^7 + z^8 + z^9 + O(z^10))*x
sage: parent(x/(1-z))
Power Series Ring in x over Laurent Series Ring in z over Rational Field 

sage: parent(x/(1-z))
Power Series Ring in x over Laurent Series Ring in z over Rational Field

Oh, I'm not sure if the change to Laurent Series Ring should be welcomed here.

Yes, the change to Laurent series ring is as advertised: inverting an element of K leads to an element in its fraction field. You can use

(1-z).inverse_of_unit()

if you want the inverse of an element you know is a unit, as an element of the ring.

@nbruin: Ok, it seems that what I really miss here is the ability to compute x//(1-z), which should end up in the same ring as x. It gives me

TypeError: unsupported operand parent(s) for //: 'Power Series Ring in z over Rational Field' and 'Power Series Ring in z over Rational Field'

There is a related issue #38570, which is however about multivariate power series, while here we have a univariate one.

There is a related issue #38570, which is however about multivariate power series, while here we have a univariate one.

That issue is about // with units from the base ring (and definitely something that can be fixed). Here it's about // with a power series itself. Over a field, k[[x]] is a euclidean ring, so // and % can definitely work. In fact their operations are already implemented:

sage: R.<z>=QQ[[]]
sage: (z).__divmod__(1-z)
(z + z^2 + z^3 + z^4 + z^5 + z^6 + z^7 + z^8 + z^9 + z^10 + z^11 + z^12 + z^13 + z^14 + z^15 + z^16 + z^17 + z^18 + z^19 + z^20 + O(z^21),
 0)
sage: (1-z).__divmod__(z)
(0, 1 - z)

they just need to be exposed under // and %.

For multivariate power series rings it's a little trickier because these are not euclidean rings. One can define "remainder" in those cases by imposing an appropriate (local) monomial ordering. Alternatively, we could just have // as a partial operation, which only produces a result if the denominator divides the numerator (which is always if the denominator is a unit) and produces an error otherwise.

Alternatively, we could just have // as a partial operation, which only produces a result if the denominator divides the numerator (which is always if the denominator is a unit) and produces an error otherwise.

This is what I have in mind for multivariate power series. I'd really appreciate it implemented.

Alternatively, we could just have // as a partial operation, which only produces a result if the denominator divides the numerator (which is always if the denominator is a unit) and produces an error otherwise.

This is what I have in mind for multivariate power series. I'd really appreciate it implemented.

Hi @maxale,
I’ve tried implementing the changes for division by handling elements in the base ring as units.
Do you think this is the right approach, or should I consider an alternative method? Would appreciate your feedback!

@vidipsingh testing for unit may be a bad idea - please see discussion in #39318 (comment) Anyway, @nbruin is an expert here and can provide much better feedback.

@vidipsingh testing for unit may be a bad idea - please see discussion in #39318 (comment) Anyway, @nbruin is an expert here and can provide much better feedback.

Hi @maxale, Thanks for the feedback! I'll check out the discussion.