I agree that the behaviour you would like is desirable in some
circumstances, but possibly not all. Note that to get the square root of
your power series, as well as adjoining t^{1/2} when the valuation is odd,
you also need to adjoin the square root of the leading coefficient (if it
is not a square). In your example the square roots are power series in
t^{1/2} with coefficients in QQ(\sqrt{2}). So you could argue that you
would be making a degree 4 extension to get a square root, which is
overkill. (In an analogous way perhaps some people might extend Q to get
(-2).sqrt() by adjoinging I=sqrt(-1) and sqrt(2), and say that the answer
is sqrt(2)*I. I know that I would not give that as the answer, but I bet
many people who are not number theorists would "simplify" sqrt(-2) like
that).
Perhaps the simplest quadratic extension for your example would be QQ[[s]]
with s^2=2*t. How about that as a compromise? i.e. if the lowest order
term is not an n'th power then adjoin its n'th root?
John
On 21 December 2017 at 11:17, Vincent Delecroix <[email protected]>
wrote:
> Dear all,
>
> I might introduce a non backward incompatible change for square root (and
> more generally n-th roots) of power series.
>
> While working on [1] I stumbled on a weird implementation of square root
> for power series [2]. Namely, when extend=True it might just return a
> formal element p so that p^2 is the initial series (example at [3])
>
> sage: K.<t> = PowerSeriesRing(QQ, 5)
> sage: f = 2*t + t^3 + O(t^4)
> sage: s = f.sqrt(extend=True, name='sqrtf'); s
> sqrtf
> sage: s^2
> 2*t + t^3 + O(t^4)
>
> A much more meaningful answer is to return a Puiseux series in t^(1/2)
> which is the natural algebraic closure of power series (at least in
> characteristic zero). Of course, I am aware that Puiseux series are not yet
> there... and this is one of our oldest open ticket [4].
>
> Anybody disagree?
>
> [1] https://trac.sagemath.org/ticket/10720
>
> [2] https://github.com/sagemath/sagetrac-mirror/blob/master/src/
> sage/rings/power_series_ring_element.pyx#L1186
>
> [3] https://github.com/sagemath/sagetrac-mirror/blob/master/src/
> sage/rings/power_series_ring_element.pyx#L1253
>
> [4] https://trac.sagemath.org/ticket/4618
>
> Vincent
>
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