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SlackIdeals :: rehomogenizePolynomial

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

Synopsis

Description

Given the symbolic slack matrix X of a polytope, we may set to 1 the variables corresponding of a spanning forest of the bipartite graph associated with X. It can be useful to rehomogenize a polynomial in the variables that are left, reversing the dehomogenization process.

This function produces the rehomogenization of a polynomial following the spanning forest backwards. It is useful in the computation of the rehomogenization of the generators of an ideal, for example of the slack ideal.

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    1 2 3 4 6 7 9 12

o5 : R

See also

Ways to use rehomogenizePolynomial :