slackIdeal -- computes the slack ideal

Synopsis

• Usage:

  I = slackIdeal(d, V)

  I = slackIdeal M

  I = slackIdeal P

  I = slackIdeal C

  I = slackIdeal V

  I = slackIdeal(d, S)

  I = slackIdeal S

• Inputs:
  • d, an integer, the dimension of the polytope or 1 less than the rank of the matroid
  • V, a list, a list of vertices of a polytope, the vectors of a linear matroid, or facet sets of an abstract polytope
  • M, a matroid, a matroid
  • P, a convex polyhedron, a polytope
  • C, a convex rational cone, a cone
  • S, a matrix, a (symbolic) slack matrix
• Optional inputs:
  • CoefficientRing => ..., default value QQ, specifies the coefficient ring of the underlying ring of the ideal
  • Object => ..., default value "polytope", specify combinatorial object
  • Saturate => ..., default value "all", specifies saturation strategy to be used
  • Strategy => ..., default value Eliminate, specifies saturation strategy to be used
  • Vars => ..., default value {}, specifies the variables to use to create the underlying ring of the ideal
• Outputs:
  • I, an ideal, the slack ideal - the saturated ideal of (d+2)-minors of the symbolic slack matrix

Description

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 ]
                  0   7

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 ]
                  0   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  ]
                  0   11

See also

• slackMatrix -- computes the slack matrix of a given realization
• symbolicSlackMatrix -- computes the symbolic slack matrix