latticePoints -- computes the lattice points of a polytope

Usage:

L = latticePoints P
Inputs:
- P, a convex polyhedron, which must be compact
Outputs:
- L, a list, containing the lattice points as matrices over ZZ with only one column

Description

latticePoints can only be applied to polytopes, i.e. compact polyhedra. It embeds the polytope on height 1 in a space of dimension plus 1 and takes the Cone over this polytope. Then it projects the elements of height 1 of the Hilbert basis back again.

i1 : P = crossPolytope 3

o1 = {ambient dimension => 3           }
      dimension of lineality space => 0
      dimension of polyhedron => 3
      number of facets => 8
      number of rays => 0
      number of vertices => 6

o1 : Polyhedron

i2 : latticePoints P

o2 = {| -1 |, | 0  |, | 0  |, 0, | 0 |, | 0 |, | 1 |}
      | 0  |  | -1 |  | 0  |     | 0 |  | 1 |  | 0 |
      | 0  |  | 0  |  | -1 |     | 1 |  | 0 |  | 0 |

o2 : List

i3 : Q = cyclicPolytope(2,4)

o3 = {ambient dimension => 2           }
      dimension of lineality space => 0
      dimension of polyhedron => 2
      number of facets => 4
      number of rays => 0
      number of vertices => 4

o3 : Polyhedron

i4 : latticePoints Q

o4 = {0, | 1 |, | 1 |, | 1 |, | 2 |, | 2 |, | 2 |, | 3 |}
         | 1 |  | 2 |  | 3 |  | 4 |  | 5 |  | 6 |  | 9 |

o4 : List

Ways to use latticePoints:

latticePoints(Polyhedron)

For the programmer

The object latticePoints is a method function.

The source of this document is in OldPolyhedra.m2:6341:0.