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# regularSubdivision -- Computes the regular cell decomposition

- Usage:
    `L = regularSubdivision(P,w)`
    `L = regularSubdivision(M,w)`
- Inputs:
  - P, a convex polyhedron,
  - w, a matrix, Row matrix containing the weights.
  - M, a matrix, Matrix containing the points.
- Outputs:
  - L, a list, List of polyhedra or List of lists of indices indicated which points form a cell.

## Description

This function computes the regular subdivision of P given by the weight vector w. This is computed by placing the i-th lattice point of P on height w_i in n+1 space, taking the convexHull of these with the ray (0,...,0,1), and projecting the compact faces into n space. Note that the polyhedron must be compact, i.e. a polytope and the length of the weight vector must be the number of lattice points.

This function can also be used to compute the regular subdivision given a matrix M of points and a weight vector w. The points are lifted to the weights given by the matrix w, and the lower envelope is computed.

    i1 : P = crossPolytope 3

    o1 = P

    o1 : Polyhedron

    i2 : w =  matrix {{1,2,2,2,2,2,1}}

    o2 = | 1 2 2 2 2 2 1 |

                  1       7
    o2 : Matrix ZZ  <-- ZZ

    i3 : L = regularSubdivision(P,w)

    o3 = {Polyhedron{...1...}, Polyhedron{...1...}, Polyhedron{...1...},
         ------------------------------------------------------------------------
         Polyhedron{...1...}}

    o3 : List

    i4 : apply(L,vertices)

    o4 = {| -1 1 0  0  |, | -1 1 0  0 |, | -1 1 0 0  |, | -1 1 0 0 |}
          | 0  0 -1 0  |  | 0  0 -1 0 |  | 0  0 1 0  |  | 0  0 1 0 |
          | 0  0 0  -1 |  | 0  0 0  1 |  | 0  0 0 -1 |  | 0  0 0 1 |

    o4 : List

    i5 : M = matrix {{1,0,1,0},{1,1,0,0}};

                  2       4
    o5 : Matrix ZZ  <-- ZZ

    i6 : w = matrix {{1,0,0,1}};

                  1       4
    o6 : Matrix ZZ  <-- ZZ

    i7 : S = regularSubdivision (M,w)

    o7 = {{0, 1, 2}, {1, 2, 3}}

    o7 : List

## Ways to use regularSubdivision:

- `regularSubdivision(Matrix,Matrix)`
- `regularSubdivision(Polyhedron,Matrix)`

## For the programmer

The object regularSubdivision is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Polyhedra/documentation/documentation.m2:376:0.