rehomogenizePolynomial -- rehomogenization of a polynomial reversing the dehomogenization of the slack matrix

Synopsis

• Usage:

  h = rehomogenizePolynomial X

  h = rehomogenizePolynomial(X, Y, T, g)

• Inputs:
  • X, a matrix, a symbolic slack matrix X
  • Y, a matrix, dehomogenized slack matrix Y
  • T, a graph, a spanning forest of the bipartite graph associated to X
  • g, a ring element, a polynomial
• Outputs:
  • h, a ring element, the rehomogenization of g reversing the dehomogenization of the slack matrix

Description

i1 : R = QQ[x_1..x_12];

i2 : X = matrix {{0, x_1, 0, 0, x_2}, {x_3, 0, 0, 0, x_4}, {0, x_5, x_6, 0, 0}, {x_7, 0, x_8, 0, 0}, {0, x_9, 0, x_10, 0}, {x_11, 0, 0, x_12, 0}};

             6      5
o2 : Matrix R  <-- R

i3 : (Y, T) = setOnesForest X;

i4 : remVars := flatten entries Y - set{0_(ring Y), 1_(ring Y)};

i5 : h = rehomogenizePolynomial(X, Y, T, remVars_0^2+remVars_0*remVars_1-1)

        2 2 2 2          2 2 2 2          2 2
o5 = - x x x x x  x   + x x x x x  x   + x x x x x x x x
        1 4 6 7 10 11    2 3 5 8 10 11    2 3 5 6 7 8 9 12

o5 : R

See also

• setOnesForest -- sets to 1 variables in a symbolic slack matrix which corresponding to edges of a spanning forest
• slackIdeal -- computes the slack ideal
• symbolicSlackMatrix -- computes the symbolic slack matrix