#12995: A first step towards linear systems of hypersurfaces in Sage
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Reporter: minz | Owner: minz
Type: enhancement | Status: needs_review
Priority: minor | Milestone: sage-5.4
Component: algebraic geometry | Resolution:
Keywords: | Work issues:
Report Upstream: N/A | Reviewers: David Eklund
Authors: Moritz Minzlaff | Merged in:
Dependencies: | Stopgaps:
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Description changed by davideklund:
Old description:
> In Magma, one can do the following:
> {{{
> > Q := RationalField();
> > P<x,y,z> := ProjectiveSpace(Q,2);
> > L := LinearSystem(P,2);
> > L;
> Linear system on Projective Space of dimension 2
> Variables : x, y, z
> with 6 sections: x^2 x*y x*z y^2 y*z z^2
> > p := P ! [3,2,1];
> > L1 := LinearSystem(L,p);
> > L1;
> Linear system on Projective Space of dimension 2
> Variables : x, y, z
> with 5 sections:
> x^2 - 9*z^2
> x*y - 6*z^2
> x*z - 3*z^2
> y^2 - 4*z^2
> y*z - 2*z^2
> }}}
> Sage does not have this functionality. This patch will be a first step
> towards adding a class LinearSystem to Sage.
>
> The goal is to add a method _linear_system_as_kernel to projective spaces
> that returns a matrix whose kernel can be identified with the degree d
> hypersurfaces with multiplicity at least m at pt.
>
> (I actually need this method in the context of equimultiple liftings of
> plane curves over finite fields for which I will open a separate ticket.)
>
> '''Apply:'''
> [attachment:trac_12995_initial.patch]
> [attachment:trac_12995_review.patch]
New description:
In Magma, one can do the following:
{{{
> Q := RationalField();
> P<x,y,z> := ProjectiveSpace(Q,2);
> L := LinearSystem(P,2);
> L;
Linear system on Projective Space of dimension 2
Variables : x, y, z
with 6 sections: x^2 x*y x*z y^2 y*z z^2
> p := P ! [3,2,1];
> L1 := LinearSystem(L,p);
> L1;
Linear system on Projective Space of dimension 2
Variables : x, y, z
with 5 sections:
x^2 - 9*z^2
x*y - 6*z^2
x*z - 3*z^2
y^2 - 4*z^2
y*z - 2*z^2
}}}
Sage does not have this functionality. This patch will be a first step
towards adding a class LinearSystem to Sage.
The goal is to add a method _linear_system_as_kernel to projective spaces
that returns a matrix whose kernel can be identified with the degree d
hypersurfaces with multiplicity at least m at pt.
(I actually need this method in the context of equimultiple liftings of
plane curves over finite fields for which I will open a separate ticket.)
'''Apply:'''
* [attachment:trac_12995_initial.patch]
* [attachment:trac_12995_review.patch]
--
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/12995#comment:10>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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