next | previous | forward | backward | up | index | toc

# rank(CentralArrangement) -- compute the rank of a central hyperplane arrangement

## Synopsis

• Function: rank
• Usage:
rank A
• Inputs:
• Outputs:
• an integer, the codimension of the intersection of the defining equations

## Description

The center of a hyperplane arrangement is the intersection of its defining affine-linear equations. The rank of a hyperplane arrangement is the codimension of its center.

We illustrate this method with some basic examples.

 i1 : R = QQ[x,y,z]; i2 : B = arrangement("braid", R) o2 = {x, y, z, x - y, x - z, y - z} o2 : Hyperplane Arrangement  i3 : rank B o3 = 3 i4 : assert(rank B === rank matroid B) i5 : rank typeA 4 o5 = 4 i6 : M = arrangement("MacLane") o6 = {x , x , x , x - x , x - x , x - 6420x , x - 6420x - x , x - 6420x + 6419x } 1 2 3 1 2 1 3 2 3 1 2 3 1 2 3 o6 : Hyperplane Arrangement  i7 : rank M o7 = 3

The trivial arrangement has no equations.

 i8 : trivial = arrangement(map(R^(numgens R),R^0,0),R) o8 = {} o8 : Hyperplane Arrangement  i9 : rank trivial o9 = 0 i10 : assert(rank trivial === 0)