ehrhart -- calculates the Ehrhart polynomial of a polytope

Synopsis

• Usage:

  f = ehrhart P

• Inputs:
  • P, a convex polyhedron, which must be compact
• Outputs:
  • f, a ring element, Ehrhart polynomial as element of QQ[x]

Description

ehrhart can only be applied to polytopes, i.e. compact polyhedra. To calculate the Ehrhart polynomial, the number of lattice points in the first n dilations of the polytope are calculated, where n is the dimension of the polytope. A system of linear equations is then solved to find the polynomial.

i1 : P = convexHull transpose matrix {{0,0,0},{1,0,0},{0,1,0},{1,1,3}}

o1 = {ambient dimension => 3           }
      dimension of lineality space => 0
      dimension of polyhedron => 3
      number of facets => 4
      number of rays => 0
      number of vertices => 4

o1 : Polyhedron

i2 : ehrhart P

     1 3    2   3
o2 = -x  + x  + -x + 1
     2          2

o2 : QQ[x]