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))
The documentation for the ehrhart polynomial
method
gives hints about all that.
To access it in text mode, use a question mark:
sage: P.ehrhart_polynomial?
The documentation can also be browsed rendered in html:
sage: browse_sage_doc(P.ehrhart_polynomial)
To access the corresponding source code:
sage: P.ehrhart_polynomial??
To discover the integral_points_count
and integral_points
methods,
use tab-completion:
sage: P.int<TAB>
where <TAB>
means "press the TAB key on the keyboard".