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# circuits(CentralArrangement) -- list the circuits of an arrangement

## Synopsis

• Function: circuits
• Usage:
circuits(A)
• Inputs:
• A, , hyperplane arrangement
• Outputs:
• a list, a list of circuits of $A$, each one expressed as a list of indices

## Description

A circuit is a minimal dependent set. More precisely, let $f_0,\ldots,f_{n-1}$ be the polynomials defining the hyperplanes of $A$. A circuit of $A$ is a subset $C\subseteq \{0,\ldots,n-1\}$ minimal among those for which $\{f_i : i\in C\}$ is linearly dependent.

If $M$ is the matroid of $A$, then a circuit of $A$ is the same as a circuit of $M$. In fact, circuits(A) is defined as toList \ circuits matroid A.

 i1 : A = typeA 3 o1 = {x - x , x - x , x - x , x - x , x - x , x - x } 1 2 1 3 1 4 2 3 2 4 3 4 o1 : Hyperplane Arrangement  i2 : circuits A o2 = {{0, 1, 3}, {4, 0, 2}, {1, 2, 3, 4}, {5, 1, 2}, {0, 2, 3, 5}, {0, 1, 4, ------------------------------------------------------------------------ 5}, {4, 5, 3}} o2 : List i3 : circuits matroid A o3 = {set {0, 1, 3}, set {4, 0, 2}, set {1, 2, 3, 4}, set {5, 1, 2}, set {0, ------------------------------------------------------------------------ 2, 3, 5}, set {0, 1, 4, 5}, set {4, 5, 3}} o3 : List