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circuits -- list the circuits of an arrangement

Synopsis

Usage:
circuits(A)
Inputs:
- A, a hyperplane arrangement, hyperplane arrangement
Outputs:
- L, a list, A list of circuits of A each one expressed as a list of indices.

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

flats -- list the flats of an arrangement of given rank

Ways to use `circuits` :

circuits(Arrangement)