#7644: generic power series reversion
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Reporter: was | Owner: AlexGhitza
Type: enhancement | Status: new
Priority: major | Milestone: sage-4.3.2
Component: algebra | Keywords:
Author: | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Comment(by fwclarke):
Replying to [comment:4 AlexGhitza]:
> What should we do with power series with coefficients in, say, ZZ?
Raise an error, or return a power series over the fraction field?
In general, return a power series over the fraction field. But if the
leading coefficient is a unit, then despite the fact that Lagrange
inversion involves division, the inverse series has coefficients in the
same ring as the original series. E.g., with the function defined in
[comment:3 was],
{{{
sage: PS.<t> = ZZ[[]]
sage: f = t + t^2 + O(t^10)
sage: ps_inverse_Lagrange(f)
t - t^2 + 2*t^3 - 5*t^4 + 14*t^5 - 42*t^6 + 132*t^7 - 429*t^8 + 1430*t^9 +
O(t^10)
}}}
though
{{{
sage: ps_inverse_Lagrange(f).parent()
Power Series Ring in t over Rational Field
}}}
Over a ring of finite characteristic, to use Lagrange inversion, you have
to lift to a ring of characteristic zero, invert, and then project down to
the original ring.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/7644#comment:5>
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