Internal points of lattice polytope

[Soli vishkautsan]

Is there a quick way to calculate the number of *internal* lattice points of a lattice polytope? LatticePolytope object has npoints method, which returns the total number of lattice points, including the border points.

[Volker Braun]

Not easily. There is this internal method whose output includes which points saturate which equation. Though you probably should just iterate over all points and pick the ones you like.

sage: from sage.geometry.polyhedron.ppl_lattice_polytope import LatticePolytope_PPL

sage: P =  LatticePolytope_PPL([0,0], [3,0], [0,3])

sage: P.constraints()

Constraint_System {x0>=0, x1>=0, -x0-x1+3>=0}

sage: P._integral_points_saturating()

(((0, 0), frozenset([0, 1])),

 ((0, 1), frozenset([0])),

 ((0, 2), frozenset([0])),

 ((0, 3), frozenset([0, 2])),

 ((1, 0), frozenset([1])),

 ((1, 1), frozenset([])),

 ((1, 2), frozenset([2])),

 ((2, 0), frozenset([1])),

 ((2, 1), frozenset([2])),

 ((3, 0), frozenset([1, 2])))

[Dima Pasechnik]

probably LattE (see https://www.math.ucdavis.edu/~latte/)
can do this reasonably fast. LattE can be installed as
an experimental Sage package
http://www.sagemath.org/packages/experimental/latte_int-1.6.spkg
(but there is no interface between Sage and it created yet...)