Implement conversion from laurent series to rational function field

[user202729]

As in the title. I think this behavior makes sense because QQ(RR(…)) currently uses .simplest_rational() instead of like exact value (like x / 2^53 for some integer x), so it makes sense for similar conversions to compute a good approximation too.

I cannot prove that this round-trips, however.

Note that currently conversion from power series QQ[[x]] to polynomial ring QQ[x] truncates, because of implementation detail this leads to conversion from QQ[[x]] to Frac(QQ[x]) also truncates. I think this is undesirable behavior because identical-looking elements in QQ[[x]] versus Frac(QQ[[x]]) = LaurentSeriesRing(QQ, "x") has different conversion behavior.

Elements which are already Laurent polynomial are preserved.

Side note: .one() is faster because it's cached

sage: R.<x> = QQ[]
sage: S = Frac(R)
sage: %timeit S.one()  # fast because it's cached
90.6 ns ± 1.27 ns per loop (mean ± std. dev. of 7 runs, 10,000,000 loops each)
sage: ring_one = S.ring().one()
sage: %timeit S._element_class(S, ring_one, ring_one, coerce=False, reduce=False)
3.42 μs ± 75 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)

Maybe this doesn't need to be explicitly written in the documentation (users trying to do the conversion will just see the result)

There exists also :meth:`.PowerSeries_poly.pade` which should probably be mentioned in the documentation.

Reverse direction (that one is a morphism): #39365

Possible caveat:

sage: S(Frac(V)(1/(x+1))/(x^10))
1/(x^11 + x^10)
sage: S(Frac(V)(1/(x+1))/(x^30))
(-x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)/x^30

It could be argued the latter would be simpler as 1/(x^31+x^30), but then that way gives higher denominator degree.

📝 Checklist

• The title is concise and informative.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation and checked the documentation preview.

⌛ Dependencies

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[tscrim]

Thank you. LGTM.

[user202729]

This is not perfect yet. Ideally it should change_ring the Laurent series to the target field before doing the approximation.

R.<x> = QQ[]
T.<t> = R[[]]
f = 1 + (x + 1)*t + (x^2 + 2*x + 1)*t^2 + (x^3 + 3*x^2 + 3*x + 1)*t^3 + O(t^4)  # belong to R[[t]], not Frac(R)[[t]]
Frac(Frac(R)[t])( f )  # error!
Frac(Frac(R)[t])( Frac(R)[[t]]( f ) )  # work

We want the # error! line to work just like the last line.