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HyperplaneArrangements :: circuits

circuits -- list the circuits of an arrangement

Synopsis

Description

By definition, a circuit is a minimal set of hyperplanes with linearly dependent normal vectors.
i1 : R := QQ[x,y,z];
i2 : A := arrangement {x,y,z,x-y,x-z,y-z};
i3 : L := circuits A

o3 = {{3, 4, 5}, {1, 2, 5}, {0, 2, 4}, {0, 1, 3}, {0, 1, 4, 5}, {0, 2, 3, 5},
     ------------------------------------------------------------------------
     {1, 2, 3, 4}}

o3 : List
i4 : (C -> (tolist A)_C)\L

o4 = {{x - y, x - z, y - z}, {y, z, y - z}, {x, z, x - z}, {x, y, x - y}, {x,
     ------------------------------------------------------------------------
     y, x - z, y - z}, {x, z, x - y, y - z}, {y, z, x - y, x - z}}

o4 : List
An arrangement has circuits of length 2 if and only if it has repeated hyperplanes:
i5 : A' := restriction(A,x)

o5 = {y, z, -y, -z, y - z}

o5 : Hyperplane Arrangement 
i6 : circuits A'

o6 = {{1, 3}, {0, 2}, {2, 3, 4}, {0, 3, 4}, {1, 2, 4}, {0, 1, 4}}

o6 : List

See also

Ways to use circuits :

For the programmer

The object circuits is a method function.