restriction -- restriction of arrangement to flat/hyperplane

Synopsis

Usage:

restriction(A,F) or restriction(A,x) or restriction(A,I)
Inputs:
- A, a hyperplane arrangement, a hyperplane arrangement (optional)
- F, an intersection of hyperplane(s), flat to which you restrict
- x, a ring element, equation of hyperplane to which you restrict
- I, an ideal, an ideal defining a subspace to which you restrict
Outputs:
- a hyperplane arrangement, the restriction of A

Description

The restriction of an arrangement to the subspace X indexed by a flat is the (multi)set of hyperplanes H intersect X for all H in the arrangement A. In the first case, one can also write A^F.

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 : L := flats(2,A)

o2 = {{0, 1, 3}, {0, 2, 4}, {0, 5}, {1, 4}, {1, 2, 5}, {2, 3}, {3, 4, 5}}

o2 : List

i3 : A' := restriction first L

o3 = {x  - x , x  - x , x  - x }
       3    4   3    4   3    4

o3 : Hyperplane Arrangement

i4 : x := (ring A)_0  -- the subspace need not be in the arrangement

o4 = x
      1

o4 : QQ[x ..x ]
         1   4

i5 : restriction(A,x)

o5 = {-x , -x , -x , x  - x , x  - x , x  - x }
        2    3    4   2    3   2    4   3    4

o5 : Hyperplane Arrangement

The restriction is, in general, a multiarrangement. Use trim to eliminate repeated hyperplanes. For example,

i6 : trim A'

o6 = {x  - x }
       3    4

o6 : Hyperplane Arrangement

Ways to use `restriction` :

"restriction(Arrangement,Flat)"
"restriction(Arrangement,Ideal)"
"restriction(Arrangement,RingElement)"
"restriction(Flat)"
restriction(Arrangement,ZZ) (missing documentation)

For the programmer

The object restriction is a method function.

restriction -- restriction of arrangement to flat/hyperplane

Synopsis

Description

See also

Ways to use restriction :

For the programmer

Ways to use `restriction` :