#11800: Problem with points at infinity in hyperelliptic curves
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Reporter: gaudry | Owner: AlexGhitza
Type: defect | Status: needs_work
Priority: minor | Milestone: sage-5.0
Component: algebraic geometry | Keywords: ecc2011, sd35,
hyperelliptic curve, conic
Work_issues: | Upstream: N/A
Reviewer: Marco Streng | Author: David Eklund
Merged: | Dependencies: #11930
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Comment(by davideklund):
Replying to [comment:18 mstreng]:
> I see that the new patch isn't set to "needs review" yet, but I'll
comment on it anyway: I think some explanation of the situation at
infinity is in order in the documentation. How about the following?
>
> "This method currently lists points in the plane projective model, which
means that one point (0:1:0) at infinity is returned if the degree of the
curve is at least 4. Later implementations may consider the more
mathematically correct desingularisation at infinity, replacing (0:1:0) by
0 or 2 smooth rational points if the degree is even." + EXAMPLE of degree
6.
>
> Also, your method of finding points at infinity is linear in the field
order, this can be improved by taking (1:y:0) and solving for y (i.e.,
extracting 1 square root), exactly as currently the affine points are
found by taking (x:y:1) for each x and solving for y.
Thank you for your comments! I hoped to attend to this earlier. I think I
will get some time to work on it soon.
Yes, some comment like the one you supply would be in order in the
documentation. About the method for finding points: I used brute force
since I deliberately choose simple code before efficiency. But I will
implement the refined approach you suggest.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/11800#comment:19>
Sage <http://www.sagemath.org>
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