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SlackIdeals :: slackIdeal

slackIdeal -- computes the slack ideal

Synopsis

Description

The slack ideal of a d-polytope or rank d+1 matroid is the ideal of (d+2)-minors of its symbolic slack matrix, saturated by the product of the variables in the matrix.

i1 : V = {{0, 0}, {1, 0}, {1, 1}, {0, 1}};
i2 : I = slackIdeal V

Order of vertices is 
{{0, 0}, {1, 0}, {0, 1}, {1, 1}}

o2 = ideal(x x x x  - x x x x )
            0 3 5 6    1 2 4 7

o2 : Ideal of QQ[x , x , x , x , x , x , x , x ]
                  0   1   2   3   4   5   6   7

If a list of points is given it can be treated as the vertices of a polytope, the ground set of a matroid or the facets of an abstract polytope by specifying the option Object. The default is as a polytope.

i3 : V = {{0, 0}, {1, 0}, {1, 1}, {0, 1}};
i4 : IP = slackIdeal V

Order of vertices is 
{{0, 0}, {1, 0}, {0, 1}, {1, 1}}

o4 = ideal(x x x x  - x x x x )
            0 3 5 6    1 2 4 7

o4 : Ideal of QQ[x , x , x , x , x , x , x , x ]
                  0   1   2   3   4   5   6   7
i5 : IM = slackIdeal(V, Object => "matroid")

o5 = ideal (x x x   + x x x  , x x x  + x x x  , x x x  + x x x  , x x x  +
             4 8 10    5 7 11   1 8 9    2 6 11   0 5 9    2 3 10   0 4 6  
     ------------------------------------------------------------------------
     x x x , x x x x   - x x x x  , x x x x  - x x x x  , x x x x  -
      1 3 7   1 3 8 10    0 5 6 11   0 4 8 9    2 3 7 11   1 5 7 9  
     ------------------------------------------------------------------------
     x x x x  )
      2 4 6 10

o5 : Ideal of QQ[x , x , x , x , x , x , x , x , x , x , x  , x  ]
                  0   1   2   3   4   5   6   7   8   9   10   11

See also

Ways to use slackIdeal :