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# rehomogenizeIdeal -- rehomogenization of a the dehomogenized slack ideal

## Synopsis

• Usage:
H = rehomogenizeIdeal(d, X)
H = rehomogenizeIdeal(d, Y, T)
• Inputs:
• d, an integer, dimension of the polytope
• X, , a symbolic slack matrix X of a d-polytope
• Y, , dehomogenized symbolic slack matrix Y of d-polytope
• T, , spanning forest whose edges match the ones in Y
• Optional inputs:
• Saturate => ..., default value "all", specifies saturation strategy to be used
• Strategy => ..., default value Eliminate, specifies saturation strategy to be used
• Outputs:
• H, an ideal, the rehomogenization of 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

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

## Ways to use rehomogenizeIdeal :

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

## For the programmer

The object rehomogenizeIdeal is .