implement the projective dual of a plane curve

[Alex Ghitza]

This was requested by Ursula Whitcher on sage-support. She adds:

  I'm playing with a family of plane curves with rational coefficients in 
  the complex projective plane.  So rational or complex numbers would be 
  enough for me to test examples.  In a perfect world I'd be able to 
  specify a family using rational functions of arbitrary constants 
  (something like a x^2 + b/(a-1) y^2), and compute the projective dual in 
  terms of those constants.

A good place to start in implementing this could be ​http://www.emilvolcheck.com/dual.ps

[David Eklund]

The attached patch implements this for (reduced and irreducible) hypersurfaces over the rationals. I intend to generalize this.

I use Grobner bases and elimination. Resultants might be faster so I might switch to that approach.

If you plug in a variety I think the dual should be reduced. I'm not sure exactly what the scheme structure of the output is at the moment.

Something related to think about for the general case: given a subscheme X of projective space, what should the dual of X be?

I will look at the approach described in the attached paper. When the dual is a hypersurface and has smaller degree than "expected", that approach seems to be better than the one used at the moment.

[David Eklund]

The patch implements duals of projective hypersurfaces. Patch rebased on top of Sage 5.2.

[David Eklund]

Patch rebased on top of Sage 5.2.

[Frédéric Chapoton]

Here is a git branch. Maybe this could be considered as need review ?

---

New commits:

​50eb935	Trac 8801: Add duals of projective hypersurfaces.
​04c7bd0	trac #8801 make sure that tests pass

[David Eklund]

I guess it could be reviewed. It could be useful as it is. I had some plans to do something better but someone stole all my time... There is no particular reason to stick to hypersurfaces, it was just meant as an initial simplification.

[David Eklund]

Nor is there any particular reason to stick to varieties over Q.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​d8cb633

Merge branch 'u/chapoton/8801' into 6.5.b6

[Frédéric Chapoton]

turning that into needs review to require feedback:

what is the class of rings over which this can currently work ?

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​4aa3fb9	Merge branch 'u/chapoton/8801' into 6.6.b1
​ba23a27	trac #8801 correct use of elim lib of singular

[Frédéric Chapoton]

nobody interested by this ticket ?

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​5a4b960

Merge branch 'u/chapoton/8801' into 7.0.rc1