#20839: Basic intersection analysis for algebraic curves
-------------------------------------+-------------------------------------
Reporter: gjorgenson | Owner:
Type: enhancement | Status: needs_review
Priority: minor | Milestone: sage-7.3
Component: algebraic | Resolution:
geometry | Merged in:
Keywords: gsoc2016 | Reviewers: Ben Hutz
Authors: Grayson Jorgenson | Work issues:
Report Upstream: N/A | Commit:
Branch: | f775a0ffaa6a969b5c3c4dc7a03862f0e71fdc58
u/gjorgenson/ticket/20839 | Stopgaps:
Dependencies: |
-------------------------------------+-------------------------------------
Changes (by bhutz):
* reviewer: => Ben Hutz
Comment:
- intersects_at(): why not but both attempts in the same try block?
- is_complete_intersection: Boolean can go on the same line at OUTPUT
not the radical ideal per our discussion today
here is another noncomplete intersection example
{{{
P.<x,y,z,w>=ProjectiveSpace(QQ,3)
X= Curve([x*z-y^2,z*(y*w-z^2) - w*(x*w-y*z)])
X.is_complete_intersection()
}}}
- intersection_multiplicity: an integer can go on the same line as OUTPUT
good these add up to 4 as bezout's theorem implies
{{{
K.<i>=QuadraticField(-1)
A.<x,y>=AffineSpace(K,2)
C = Curve([x^2-y])
D = Curve([x^2+y^2])
for t in C.intersection(D).rational_points():
C.intersection_multiplicity(D,t)
}}}
also good
{{{
K.<i>=QuadraticField(-1)
A.<x,y>=AffineSpace(K,2)
C = Curve([y^2-x^3])
D = Curve([(x^2+y^2)^2 - 4*x^2*y^2])
for t in C.intersection(D).rational_points():
C.intersection_multiplicity(D,t)
}}}
so for plane curves this looks ok. Does this really work in dimension
greater than 2? I don't think it does, I think the intersection number
possibly has lower order terms. Regardless, in looking at this, the
following example died
{{{
K.<i>=QuadraticField(-1)
A.<x,y,z>=AffineSpace(K,3)
C = Curve([x^2-y,z^2+x^2])
D = Curve([x^2+y^2,x^2+z])
for t in C.intersection(D).rational_points():
C.intersection_multiplicity(D,p)
}}}
- I think an transverse check would be nice too:
is_transverse() - Returns true if and only if the point p is a nonsingular
point of both plane curves C and D and the curves have distinct tangents
there.
--
Ticket URL: <http://trac.sagemath.org/ticket/20839#comment:4>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
--
You received this message because you are subscribed to the Google Groups
"sage-trac" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at https://groups.google.com/group/sage-trac.
For more options, visit https://groups.google.com/d/optout.
