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The Ehrhart polynomial $p$ of a polyhedron $P$ helps counting integral points in its dilates.

Indeed, for each integer $n$, the number of integral points in the dilate $n P$ is given by $p(n)$.

Sage also provides a direct method for counting integral points.

Here is an illustration.

Construct a polyhedron:

sage: vertex_list = [(1, 9, 0), (2, 4, 5), (3, 3, 2), (5, 0, 9),
....:                (6, 1, 9), (6, 3, 1), (9, 3, 2), (8, 6, 9)]
sage: P = Polyhedron(vertices=vertex_list)

Find its Ehrhart polynomial via latte (requires installing the optional package latte_int):

sage: p = P.ehrhart_polynomial(engine='latte')
sage: p
893/6*t^3 + 13/2*t^2 + 23/3*t + 1

Get the number of integral points in the polyhedron via the Ehrhart polynomial:

sage: p(1)
164

Get the number of integral points in the polyhedron directly:

sage: P.integral_points_count()
164

The Ehrhart polynomial $p$ of a polyhedron $P$ helps counting integral points in its dilates.

Indeed, for each integer $n$, the number of integral points in the dilate $n P$ is given by $p(n)$.

Sage also provides a direct method for counting integral points.

Here is an illustration.

Construct a polyhedron:

sage: vertex_list = [(1, 9, 0), [(0, 0, 1), (1, 0, 0), (1, 0, 1),
....:                (1, 1, 0), (1, 1, 1), (2, 4, 5), (3, 3, 2), (5, 0, 9),
....:                (6, 1, 9), (6, 3, 1), (9, 3, 2), (8, 6, 9)]
1, 0)]
sage: P = Polyhedron(vertices=vertex_list)

Inspect it:

sage: P
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 6 vertices

Find its Ehrhart polynomial via latte (requires installing the optional package latte_int):

sage: p = P.ehrhart_polynomial(engine='latte')
sage: p
893/6*t^3 2/3*t^3 + 13/2*t^2 2*t^2 + 23/3*t 7/3*t + 1

Get the number of integral points in the polyhedron via the Ehrhart polynomial:

sage: p(1)
164
6

Get the number of integral points in the polyhedron directly:

sage: P.integral_points_count()
164

sage: points = P.integral_points()
sage: points
((1, 0, 0), (0, 0, 1), (1, 0, 1), (1, 1, 0), (2, 1, 0), (1, 1, 1))

The Ehrhart polynomial $p$ of a polyhedron $P$ helps counting integral points in its dilates.

Indeed, for each integer $n$, the number of integral points in the dilate $n P$ is given by $p(n)$.

Sage also provides a direct method for counting integral points.

Here is an illustration.

Construct a polyhedron:

sage: vertex_list = [(0, 0, 1), (1, 0, 0), (1, 0, 1),
....:                (1, 1, 0), (1, 1, 1), (2, 1, 0)]
sage: P = Polyhedron(vertices=vertex_list)

Inspect it:

sage: P
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 6 vertices

Find its Ehrhart polynomial via latte (requires installing the optional package latte_int):

sage: p = P.ehrhart_polynomial(engine='latte')
sage: p
2/3*t^3 + 2*t^2 + 7/3*t + 1

Get the number of integral points in the polyhedron via the Ehrhart polynomial:

sage: p(1)
6

Get the number of integral points in the polyhedron directly:

sage: P.integral_points_count()
6

Besides counting integral points, one can also be interested in finding these points explicitly.

The method integral_points provides that:

sage: points = P.integral_points()
sage: points
((1, 0, 0), (0, 0, 1), (1, 0, 1), (1, 1, 0), (2, 1, 0), (1, 1, 1))

sage: P.ehrhart_polynomial?

sage: browse_sage_doc(P.ehrhart_polynomial)

sage: P.ehrhart_polynomial??

sage: P.int<TAB>