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rehomogenizeIdeal -- rehomogenization of a the dehomogenized slack ideal

Description

It computes the rehomogenization of the dehomogenized slack ideal, applying the rehomogenize function to its generators.

i1 : V = {{0, 0, 0}, {1, 0, 0}, {1, 1, 0}, {0, 1, 0}, {0, 0, 1}, {0, 1, 1}};
 
i2 : X = symbolicSlackMatrix V

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

o2 = | 0   0    x_0 0    x_1 |
     | x_2 0    x_3 0    0   |
     | 0   x_4  0   0    x_5 |
     | x_6 x_7  0   0    0   |
     | 0   0    x_8 x_9  0   |
     | 0   x_10 0   x_11 0   |

                         6                  5
o2 : Matrix (QQ[x ..x  ])  <-- (QQ[x ..x  ])
                 0   11             0   11
 
i3 : H = rehomogenizeIdeal(3, X)

o3 = ideal (x x x x   - x x x x  , x x x x   - x x x x  , x x x x  -
             3 6 9 10    2 7 8 11   0 5 9 10    1 4 8 11   1 3 4 6  
     ------------------------------------------------------------------------
     x x x x )
      0 2 5 7

o3 : Ideal of QQ[x ..x  ]
                  0   11
 
i4 : R = QQ[x_0..x_11];
 
i5 : Y = matrix {{0, 0, 1, 0, 1}, {1, 0, 1, 0, 0}, {0, x_4, 0, 0, 1}, {1, 1, 0, 0, 0}, {0, 0, 1, 1, 0}, {0, x_10, 0, 1, 0}};

             6      5
o5 : Matrix R  <-- R
 
i6 : T = graph(QQ[y_0, y_1, y_2, y_3, y_4, y_5, y_6, y_7, y_8, y_9, y_10], {{y_1, y_6}, {y_3, y_6}, {y_3, y_7}, {y_0, y_8}, {y_1, y_8}, {y_4, y_8}, {y_4, y_9}, {y_5, y_9}, {y_0, y_10}, {y_2, y_10}});
 
i7 : rehomogenizeIdeal(3, Y, T)

o7 = ideal (x x x x   - x x x x  , x x x x   - x x x x  , x x x x  -
             3 6 9 10    2 7 8 11   0 5 9 10    1 4 8 11   1 3 4 6  
     ------------------------------------------------------------------------
     x x x x )
      0 2 5 7

o7 : Ideal of R

See also

Ways to use rehomogenizeIdeal:

  • rehomogenizeIdeal(ZZ,Matrix)
  • rehomogenizeIdeal(ZZ,Matrix,Graph)

For the programmer

The object rehomogenizeIdeal is a method function with options.

The source of this document is in /__w/M2/M2/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SlackIdeals.m2:2296:0.