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

Synopsis

Function: circuits
Usage:

circuits(A)
Inputs:
- A, a central hyperplane arrangement, 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 {0, 2, 4}, set {1, 2, 3, 4}, set {1, 2, 5}, set {0,
     ------------------------------------------------------------------------
     2, 3, 5}, set {0, 1, 4, 5}, set {3, 4, 5}}

o3 : List

Ways to use this method: