arrangement -- create a hyperplane arrangement

Synopsis

• Usage:

  arrangement(L,R) or arrangement(M) or arrangement(M,R) or

  arrangement Q

• Inputs:
  • L, a list of affine-linear equations in the ring R
  • R, a polynomial ring or linear quotient of a polynomial ring
  • M, a matrix whose columns represent linear forms defining hyperplanes
  • Q, a product of linear forms
• Outputs:
  • a hyperplane arrangement, the hyperplane arrangement determined by L and R

Description

Probably the best-known hyperplane arrangement is the braid arrangement consisting of all the diagonal hyperplanes. In 4-space, it is constructed as follows:

i1 : S = ZZ[w,x,y,z];

i2 : A3 = arrangement {w-x,w-y,w-z,x-y,x-z,y-z}

o2 = {w - x, w - y, w - z, x - y, x - z, y - z}

o2 : Hyperplane Arrangement 

i3 : describe A3

o3 = {w - x, w - y, w - z, x - y, x - z, y - z}

i4 : R = S/ideal(w+x+y+z)

o4 = R

o4 : QuotientRing

i5 : A3' = arrangement({w-x,w-y,w-z,x-y,x-z,y-z},R)

o5 = {- 2x - y - z, - x - 2y - z, - x - y - 2z, x - y, x - z, y - z}

o5 : Hyperplane Arrangement 

i6 : describe A3'

o6 = {- 2x - y - z, - x - 2y - z, - x - y - 2z, x - y, x - z, y - z}

i7 : trivial = arrangement({},S)

o7 = {}

o7 : Hyperplane Arrangement 

i8 : describe trivial

o8 = {}

i9 : ring trivial

o9 = S

o9 : PolynomialRing

i10 : use S;

i11 : arrangement (x^2*y^2*(x^2-y^2)*(x^2-z^2))

o11 = {y, y, x, x, x - z, x + z, x - y, x + y}

o11 : Hyperplane Arrangement 

Caveat

If arrangement Q is used, the order of the factors is determined internally.

See also

• HyperplaneArrangements -- hyperplane arrangements
• arrangement(String,PolynomialRing) -- look up a built-in hyperplane arrangement