randomArrangement(ZZ,PolynomialRing,ZZ) -- generate an arrangement at random

Synopsis

Function: randomArrangement
Usage:

randomArrangement(n,R,N)
Inputs:
- n, an integer, number of hyperplanes
- R, a polynomial ring, a polynomial ring over which to define the arrangement, or a number of variables l instead
- N, an integer, absolute value of upper bound on coefficients
Optional inputs:
- Validate => a Boolean value, default value false, if true, the method will attempt to return an arrangement whose underlying matroid is uniform.
Outputs:
- a hyperplane arrangement, a random rational arrangement of $n$ hyperplanes defined over $R$.

Description

As $N$ increases, the random arrangement is a generic arrangement (i.e., a realization of the uniform matroid with probability tending to 1. The user can require that the arrangement generated is actually generic by using the option Validate => true.

i1 : randomArrangement(4,3,5)

o1 = {- 4x  + 3x  + x , 3x  + x , - 2x  + 2x , 3x  + x  + x }
          1     2    3    1    3      1     3    1    2    3

o1 : Hyperplane Arrangement

If an arrangement has the poincare polynomial of a generic arrangement, then it is itself generic.

i2 : tally apply(12, i -> poincare randomArrangement(6,3,5))

                       2      3
o2 = Tally{1 + 6T + 15T  + 10T  => 8}
                       2     3
           1 + 6T + 14T  + 9T  => 4

o2 : Tally

i3 : A = randomArrangement(6,3,5,Validate=>true)

o3 = {5x  + 4x  + 3x , - 3x  + x  - 5x , - 5x  - 3x  + 3x , 3x  + 4x  - 2x , - 2x  - 3x  + 5x , 5x  + 3x  - 5x }
        1     2     3      1    2     3      1     2     3    1     2     3      1     2     3    1     2     3

o3 : Hyperplane Arrangement

i4 : U = uniformMatroid(3,6);

i5 : assert areIsomorphic(U, matroid A)

Caveat

If the user specifies Validate => true and $N$ is too small, the method may not halt.

Ways to use this method: