isInCoisotropic -- test membership in a coisotropic hypersurface

Synopsis

Usage:

isInCoisotropic(L,I)
Inputs:
- L, an ideal, the ideal of a $k$-dimensional linear subspace $V(L)\subset\mathbb{P}^n$, which corresponds to a point $p_L\in\mathbb{G}(k,\mathbb{P}^n)$
- I, an ideal, the ideal of a $d$-dimensional projective variety $X=V(I)\subset\mathbb{P}^n$
Optional inputs:
- Duality => ..., default value null, whether to use dual Plücker coordinates
- SingularLocus => ..., default value null, pass the singular locus of the variety
Outputs:
- a Boolean value, whether $p_L$ belongs to the $s$-th associated subvariety $Z_s(X)\subset\mathbb{G}(k,\mathbb{P}^n)$ with $s = k+d+1-n$

Description

This is equivalent to isSubset(ideal tangentialChowForm(I,s),plucker L), but it does not compute the tangential Chow form.

i1 : use Grass(0,5,ZZ/33331,Variable=>x)

       ZZ
o1 = -----[x ..x ]
     33331  0   5

o1 : PolynomialRing

i2 : I = minors(2,matrix {{x_0,x_1,x_3,x_4},{x_1,x_2,x_4,x_5}}) -- rational normal scroll surface

               2                                                            
o2 = ideal (- x  + x x , - x x  + x x , - x x  + x x , - x x  + x x , - x x 
               1    0 2     1 3    0 4     2 3    1 4     1 4    0 5     2 4
     ------------------------------------------------------------------------
                2
     + x x , - x  + x x )
        1 5     4    3 5

                ZZ
o2 : Ideal of -----[x ..x ]
              33331  0   5

i3 : L = ideal(x_1-12385*x_2-16397*x_3-7761*x_4+827*x_5,x_0+2162*x_2-8686*x_3+2380*x_4+9482*x_5) -- linear 3-dimensional subspace

o3 = ideal (x  - 12385x  - 16397x  - 7761x  + 827x , x  + 2162x  - 8686x  +
             1         2         3        4       5   0        2        3  
     ------------------------------------------------------------------------
     2380x  + 9482x )
          4        5

                ZZ
o3 : Ideal of -----[x ..x ]
              33331  0   5

i4 : time isInCoisotropic(L,I) -- whether L belongs to Z_1(V(I))
     -- used 0.0172671 seconds

o4 = true

Ways to use isInCoisotropic :

isInCoisotropic(Ideal,Ideal)

For the programmer

The object isInCoisotropic is a method function with options.

isInCoisotropic -- test membership in a coisotropic hypersurface

Synopsis

Description

See also

Ways to use isInCoisotropic :

For the programmer