# circuits -- list the circuits of an arrangement

## Synopsis

• Usage:
circuits(A)
• Inputs:
• A, , 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