#10172: Rational Point algorithm bug
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Reporter: nkaplan | Owner: AlexGhitza
Type: defect | Status: needs_review
Priority: major | Milestone: sage-4.8
Component: algebraic geometry | Keywords:
Work_issues: | Upstream: None of the above - read
trac for reasoning.
Reviewer: | Author: Moritz Minzlaff
Merged: | Dependencies:
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Changes (by minz):
* status: needs_work => needs_review
Old description:
> When you have a curve defined over a projective space over a finite
> field, it looks like sometimes the rational point list coming from the
> Brill-Noether package in Singular is not the full list of rational
> points.
>
> {{{
> sage: S.<x,y,z> = GF(5)[]
> sage: g = x*z+z^2
> sage: G = Curve(g)
> sage: G.rational_points('enum')
> [(0 : 1 : 0), (1 : 0 : 0), (1 : 1 : 0), (2 : 1 : 0), (3 : 1 : 0), (4 : 0
> : 1), (4 : 1 : 0), (4 : 1 : 1), (4 : 2 : 1), (4 : 3 : 1), (4 : 4 : 1)]
> sage: G.rational_points('bn')
> [(0 : 1 : 0), (4 : 0 : 1), (4 : 1 : 1), (4 : 2 : 1), (4 : 3 : 1), (4 : 4
> : 1)]
> }}}
>
> It's not clear yet whether this is a Sage or a Singular problem.
New description:
When you have a curve defined over a projective space over a finite field,
it looks like sometimes the rational point list coming from the Brill-
Noether package in Singular is not the full list of rational points.
{{{
sage: S.<x,y,z> = GF(5)[]
sage: g = x*z+z^2
sage: G = Curve(g)
sage: G.rational_points('enum')
[(0 : 1 : 0), (1 : 0 : 0), (1 : 1 : 0), (2 : 1 : 0),
(3 : 1 : 0), (4 : 0 : 1), (4 : 1 : 0), (4 : 1 : 1),
(4 : 2 : 1), (4 : 3 : 1), (4 : 4 : 1)]
sage: G.rational_points('bn')
[(0 : 1 : 0), (4 : 0 : 1), (4 : 1 : 1), (4 : 2 : 1),
(4 : 3 : 1), (4 : 4 : 1)]
}}}
The problem was how Sage calls Singular's functions.
--
Comment:
We were missing so far that and how Singular dehomogenizes the defining
polynomial of the curve: Always with respect to the last variable. If the
corresponding line is a component of the curve, the output was missing
those points.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/10172#comment:3>
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