#8972: Inversion and fraction fields for power series rings
-----------------------------------------------+----------------------------
Reporter: SimonKing | Owner: AlexGhitza
Type: defect | Status: needs_review
Priority: major | Milestone: sage-4.4.4
Component: algebra | Resolution: fixed
Keywords: power series ring, fraction field | Author: Simon King
Upstream: N/A | Reviewer: Robert Bradshaw
Merged: | Work_issues:
-----------------------------------------------+----------------------------
Changes (by SimonKing):
* status: needs_work => needs_review
* work_issues: fix division if no fraction field exists; use ~ instead
of / in elliptic curve code =>
Comment:
I think the problems are now solved. With the last patch, that is to be
applied after all others, {{{sage -testall}}} works without errors (at
least in version sage-4.4.2).
'''__Changes introduced by the patch__'''
* In some cases, if the code expects a power series, I replaced {{{a/b}}}
for power series {{{a,b}}} by {{{a*~b}}}. Namely, the latter returns a
power series (not a Laurent series), if possible.
* I added a method {{{exp}}} for Laurent series, that returns the
exponential if the Laurent series happens to be a power series.
* With my previous patches, the underlying data of a Laurent series is
not necessarily a power series. Therefore {{{add_bigoh}}} (similarly
{{{_rmul_}}} and {{{_lmul_}}}) did in some cases not work. This is now
fixed and doctested.
* Some Power Series Rings still returned a ''formal'' fraction field.
This is now fixed, they return a Laurent series ring.
* If {{{a,b}}} are elements of a power series ring {{{R}}}, the rule is
now:
1. If {{{R}}} is an integral domain, then {{{a/b}}} always belongs to
{{{Frac(R)}}}. This is similar to {{{1/1}}} being rational, not integer.
2. If {{{R}}} is no integral domain, then {{{a/b}}} is an element of
{{{R}}} or of the Laurent series ring over the base ring of {{{R}}}, if
{{{b}}} is invertible. This is similar to division in {{{ZZ.quo(15)}}}.
3. If possible, {{{~b}}} is a power series (possibly over the fraction
field of the base of {{{R}}}). So, we always have {{{~b==1/b}}}, but the
parents may be different. I hope this is acceptable.
* A change in the default {{{_div_}}} method of {{{RingElement}}}:
Previously, the default for {{{a._div_(b)}}} was to return
{{{a.parent().fraction_field()(a,b)}}}. But this may be a problem (e.g.,
if the fraction field is not a formal fraction field but a Laurent series
ring). Therefore, if the old default results in an error,
{{{a.parent().fraction_field(a)/a.parent().fraction_field(b)}}} is tried.
'''__Timings__'''
Robert was worried about potential slowdowns in the Monsky-Washnitzer
code. It seems to me that my patches actually provide a considerably
speedup.
Without my patches:
{{{
sage -t "devel/sage-
powerseries/sage/schemes/elliptic_curves/monsky_washnitzer.py"
[3.9 s]
----------------------------------------------------------------------
All tests passed!
Total time for all tests: 3.9 seconds
}}}
With my patches:
{{{
sage -t "devel/sage-
powerseries/sage/schemes/elliptic_curves/monsky_washnitzer.py"
[2.3 s]
----------------------------------------------------------------------
All tests passed!
Total time for all tests: 2.3 seconds
}}}
So, the code seems to become faster by more than 33%!
Ready for review again...
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8972#comment:31>
Sage <http://www.sagemath.org>
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