# orientedCircuits -- compute the oriented circuits of an oriented matroid or point or vector configuration

## Synopsis

• Usage:
orientedCircuits om
orientedCocircuits A
• Inputs:
• om, ,
• A, ,
• Optional inputs:
• Homogenize => , default value true, Only valid in the second form.
• Outputs:

## Description

An oriented circuit is determined by a linear relationship on the columns of (the augmented matrix of) $A$, of minimal support. The circuit is the pair of lists of indices of the columns where the coefficients is positive (respectively negative).

 i1 : A = matrix { {0, -1, 2, 3, 4, -5, 6}, {0, 1, -4, 9, 16, 25, 36}, {0, 1, 8, -27, 64, 125, -216}} o1 = | 0 -1 2 3 4 -5 6 | | 0 1 -4 9 16 25 36 | | 0 1 8 -27 64 125 -216 | 3 7 o1 : Matrix ZZ <--- ZZ i2 : om = naiveChirotopeString A o2 = 7,4: ---0++-+----+-++++++--+-+++++++-+++ i3 : netList orientedCircuits om +------------+---------+ o3 = |{0} |{1, 2, 6}| +------------+---------+ |{0, 2, 4, 6}|{3} | +------------+---------+ |{0, 3} |{2, 5, 6}| +------------+---------+ |{0, 3, 5} |{1, 4} | +------------+---------+ |{0, 4} |{1, 2, 3}| +------------+---------+ |{0, 4} |{2, 3, 5}| +------------+---------+ |{0, 4} |{2, 5, 6}| +------------+---------+ |{0, 4, 6} |{1, 3} | +------------+---------+ |{0, 4, 6} |{3, 5} | +------------+---------+ |{0, 5} |{1, 2, 3}| +------------+---------+ |{0, 5} |{1, 2, 4}| +------------+---------+ |{0, 5, 6} |{1, 3} | +------------+---------+ |{0, 5, 6} |{1, 4} | +------------+---------+ |{1, 2, 4, 6}|{3} | +------------+---------+ |{1, 3} |{2, 5, 6}| +------------+---------+ |{1, 4} |{2, 3, 5}| +------------+---------+ |{1, 4} |{2, 5, 6}| +------------+---------+ |{1, 4, 6} |{3, 5} | +------------+---------+ |{2, 4, 5, 6}|{3} | +------------+---------+

Let's look at the linear relation giving rise to $\{\{0,3\}, \{ 2, 5, 6\}\}$.

 i4 : Ahomog = A || matrix{{7:1}} o4 = | 0 -1 2 3 4 -5 6 | | 0 1 -4 9 16 25 36 | | 0 1 8 -27 64 125 -216 | | 1 1 1 1 1 1 1 | 4 7 o4 : Matrix ZZ <--- ZZ i5 : Ahomog_{0,3,2,5,6} o5 = | 0 3 2 -5 6 | | 0 9 -4 25 36 | | 0 -27 8 125 -216 | | 1 1 1 1 1 | 4 5 o5 : Matrix ZZ <--- ZZ i6 : syz oo o6 = | -242 | | -440 | | 495 | | 72 | | 115 | 5 1 o6 : Matrix ZZ <--- ZZ