Generating Function of Polyhedron’s Integral Points

This module provides generating_function_of_integral_points() which computes the generating function of the integral points of a polyhedron.

The main function is accessible via sage.geometry.polyhedron.base.Polyhedron_base.generating_function_of_integral_points() as well.

Various

AUTHORS:

  • Daniel Krenn (2016, 2021)

ACKNOWLEDGEMENT:

  • Daniel Krenn is supported by the Austrian Science Fund (FWF): P 24644-N26 and by the Austrian Science Fund (FWF): P 28466-N35.

Functions

sage.geometry.polyhedron.generating_function.generating_function_of_integral_points(polyhedron, split=False, result_as_tuple=None, name=None, names=None, **kwds)[source]

Return the multivariate generating function of the integral points of the polyhedron.

To be precise, this returns

\[\sum_{(r_0,\dots,r_{d-1}) \in \mathit{polyhedron}\cap \ZZ^d} y_0^{r_0} \dots y_{d-1}^{r_{d-1}}.\]

INPUT:

  • polyhedron – an instance of Polyhedron_base (see also sage.geometry.polyhedron.constructor)

  • split – (default: False) a boolean or list

    • split=False computes the generating function directly, without any splitting.

    • When split is a list of disjoint polyhedra, then for each of these polyhedra, polyhedron is intersected with it, its generating function computed and all these generating functions are summed up.

    • split=True splits into \(d!\) disjoint polyhedra.

  • result_as_tuple – (default: None) a boolean or None

    This specifies whether the output is a (partial) factorization (result_as_tuple=False) or a sum of such (partial) factorizations (result_as_tuple=True). By default (result_as_tuple=None), this is automatically determined. If the output is a sum, it is represented as a tuple whose entries are the summands.

  • indices – (default: None) a list or tuple

    If this is None, this is automatically determined.

  • name – (default: 'y') a string

    The variable names of the Laurent polynomial ring of the output are this string followed by an integer.

  • names – list or tuple of names (strings), or a comma separated string

    name is extracted from names, therefore names has to contain exactly one variable name, and name and``names`` cannot be specified both at the same time.

  • Factorization_sort (default: False) and Factorization_simplify (default: True) – booleans

    These are passed on to sage.structure.factorization.Factorization when creating the result.

  • sort_factors – (default: False) a boolean

    If set, then the factors of the output are sorted such that the numerator is first and only then all factors of the denominator. It is ensured that the sorting is always the same; use this for doctesting.

OUTPUT:

The generating function as a (partial) Factorization of the result whose factors are Laurent polynomials

The result might be a tuple of such factorizations (depending on the parameter result_as_tuple) as well.

Note

At the moment, only polyhedra with nonnegative coordinates (i.e. a polyhedron in the nonnegative orthant) are handled.

EXAMPLES:

sage: from sage.geometry.polyhedron.generating_function import generating_function_of_integral_points

>>> from sage.all import *
>>> from sage.geometry.polyhedron.generating_function import generating_function_of_integral_points

from sage.geometry.polyhedron.generating_function import generating_function_of_integral_points
sage: P2 = (
....:   Polyhedron(ieqs=[(0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, -1)]),
....:   Polyhedron(ieqs=[(0, -1, 0, 1), (0, 1, 0, 0), (0, 0, 1, 0)]))
sage: generating_function_of_integral_points(P2[0], sort_factors=True)
1 * (-y0 + 1)^-1 * (-y1 + 1)^-1 * (-y0*y2 + 1)^-1
sage: generating_function_of_integral_points(P2[1], sort_factors=True)
1 * (-y1 + 1)^-1 * (-y2 + 1)^-1 * (-y0*y2 + 1)^-1
sage: (P2[0] & P2[1]).Hrepresentation()
(An equation (1, 0, -1) x + 0 == 0,
 An inequality (1, 0, 0) x + 0 >= 0,
 An inequality (0, 1, 0) x + 0 >= 0)
sage: generating_function_of_integral_points(P2[0] & P2[1], sort_factors=True)
1 * (-y1 + 1)^-1 * (-y0*y2 + 1)^-1

>>> from sage.all import *
>>> P2 = (
...   Polyhedron(ieqs=[(Integer(0), Integer(0), Integer(0), Integer(1)), (Integer(0), Integer(0), Integer(1), Integer(0)), (Integer(0), Integer(1), Integer(0), -Integer(1))]),
...   Polyhedron(ieqs=[(Integer(0), -Integer(1), Integer(0), Integer(1)), (Integer(0), Integer(1), Integer(0), Integer(0)), (Integer(0), Integer(0), Integer(1), Integer(0))]))
>>> generating_function_of_integral_points(P2[Integer(0)], sort_factors=True)
1 * (-y0 + 1)^-1 * (-y1 + 1)^-1 * (-y0*y2 + 1)^-1
>>> generating_function_of_integral_points(P2[Integer(1)], sort_factors=True)
1 * (-y1 + 1)^-1 * (-y2 + 1)^-1 * (-y0*y2 + 1)^-1
>>> (P2[Integer(0)] & P2[Integer(1)]).Hrepresentation()
(An equation (1, 0, -1) x + 0 == 0,
 An inequality (1, 0, 0) x + 0 >= 0,
 An inequality (0, 1, 0) x + 0 >= 0)
>>> generating_function_of_integral_points(P2[Integer(0)] & P2[Integer(1)], sort_factors=True)
1 * (-y1 + 1)^-1 * (-y0*y2 + 1)^-1

P2 = (
  Polyhedron(ieqs=[(0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, -1)]),
  Polyhedron(ieqs=[(0, -1, 0, 1), (0, 1, 0, 0), (0, 0, 1, 0)]))
generating_function_of_integral_points(P2[0], sort_factors=True)
generating_function_of_integral_points(P2[1], sort_factors=True)
(P2[0] & P2[1]).Hrepresentation()
generating_function_of_integral_points(P2[0] & P2[1], sort_factors=True)
>>> from sage.all import *
>>> P2 = (
...   Polyhedron(ieqs=[(Integer(0), Integer(0), Integer(0), Integer(1)), (Integer(0), Integer(0), Integer(1), Integer(0)), (Integer(0), Integer(1), Integer(0), -Integer(1))]),
...   Polyhedron(ieqs=[(Integer(0), -Integer(1), Integer(0), Integer(1)), (Integer(0), Integer(1), Integer(0), Integer(0)), (Integer(0), Integer(0), Integer(1), Integer(0))]))
>>> generating_function_of_integral_points(P2[Integer(0)], sort_factors=True)
1 * (-y0 + 1)^-1 * (-y1 + 1)^-1 * (-y0*y2 + 1)^-1
>>> generating_function_of_integral_points(P2[Integer(1)], sort_factors=True)
1 * (-y1 + 1)^-1 * (-y2 + 1)^-1 * (-y0*y2 + 1)^-1
>>> (P2[Integer(0)] & P2[Integer(1)]).Hrepresentation()
(An equation (1, 0, -1) x + 0 == 0,
 An inequality (1, 0, 0) x + 0 >= 0,
 An inequality (0, 1, 0) x + 0 >= 0)
>>> generating_function_of_integral_points(P2[Integer(0)] & P2[Integer(1)], sort_factors=True)
1 * (-y1 + 1)^-1 * (-y0*y2 + 1)^-1

P2 = (
  Polyhedron(ieqs=[(0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, -1)]),
  Polyhedron(ieqs=[(0, -1, 0, 1), (0, 1, 0, 0), (0, 0, 1, 0)]))
generating_function_of_integral_points(P2[0], sort_factors=True)
generating_function_of_integral_points(P2[1], sort_factors=True)
(P2[0] & P2[1]).Hrepresentation()
generating_function_of_integral_points(P2[0] & P2[1], sort_factors=True)
sage: P3 = (
....:   Polyhedron(
....:     ieqs=[(0, 0, 0, 0, 1), (0, 0, 0, 1, 0),
....:           (0, 0, 1, 0, -1), (-1, 1, 0, -1, -1)]),
....:   Polyhedron(
....:     ieqs=[(0, 0, -1, 0, 1), (0, 1, 0, 0, -1),
....:           (0, 0, 0, 1, 0), (0, 0, 1, 0, 0), (-1, 1, -1, -1, 0)]),
....:   Polyhedron(
....:     ieqs=[(1, -1, 0, 1, 1), (1, -1, 1, 1, 0),
....:           (0, 0, 0, 0, 1), (0, 0, 0, 1, 0), (0, 0, 1, 0, 0),
....:           (1, 0, 1, 1, -1), (0, 1, 0, 0, 0), (1, 1, 1, 0, -1)]),
....:   Polyhedron(
....:     ieqs=[(0, 1, 0, -1, 0), (0, -1, 0, 0, 1),
....:           (-1, 0, -1, -1, 1), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0)]),
....:   Polyhedron(
....:     ieqs=[(0, 1, 0, 0, 0), (0, 0, 1, 0, 0),
....:           (-1, -1, -1, 0, 1), (0, -1, 0, 1, 0)]))
sage: def intersect(I):
....:     I = iter(I)
....:     result = next(I)
....:     for i in I:
....:         result &= i
....:     return result
sage: for J in subsets(range(len(P3))):
....:     if not J:
....:         continue
....:     P = intersect([P3[j] for j in J])
....:     print('{}: {}'.format(J, P.Hrepresentation()))
....:     print(generating_function_of_integral_points(P, sort_factors=True))
[0]: (An inequality (0, 0, 0, 1) x + 0 >= 0,
      An inequality (0, 0, 1, 0) x + 0 >= 0,
      An inequality (0, 1, 0, -1) x + 0 >= 0,
      An inequality (1, 0, -1, -1) x - 1 >= 0)
y0 * (-y0 + 1)^-1 * (-y1 + 1)^-1 * (-y0*y2 + 1)^-1 * (-y0*y1*y3 + 1)^-1
[1]: (An inequality (0, -1, 0, 1) x + 0 >= 0,
      An inequality (0, 0, 1, 0) x + 0 >= 0,
      An inequality (0, 1, 0, 0) x + 0 >= 0,
      An inequality (1, -1, -1, 0) x - 1 >= 0,
      An inequality (1, 0, 0, -1) x + 0 >= 0)
(-y0^2*y2*y3 - y0^2*y3 + y0*y3 + y0) *
(-y0 + 1)^-1 * (-y0*y2 + 1)^-1 * (-y0*y3 + 1)^-1 *
(-y0*y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 1]: (An equation (0, 1, 0, -1) x + 0 == 0,
         An inequality (1, -1, -1, 0) x - 1 >= 0,
         An inequality (0, 1, 0, 0) x + 0 >= 0,
         An inequality (0, 0, 1, 0) x + 0 >= 0)
y0 * (-y0 + 1)^-1 * (-y0*y2 + 1)^-1 * (-y0*y1*y3 + 1)^-1
[2]: (An inequality (-1, 0, 1, 1) x + 1 >= 0,
     An inequality (-1, 1, 1, 0) x + 1 >= 0,
      An inequality (0, 0, 0, 1) x + 0 >= 0,
      An inequality (0, 0, 1, 0) x + 0 >= 0,
      An inequality (0, 1, 0, 0) x + 0 >= 0,
      An inequality (0, 1, 1, -1) x + 1 >= 0,
      An inequality (1, 0, 0, 0) x + 0 >= 0,
      An inequality (1, 1, 0, -1) x + 1 >= 0)
(y0^2*y1*y2*y3^2 + y0^2*y2^2*y3 + y0*y1^2*y3^2 - y0^2*y2*y3 +
 y0*y1*y2*y3 - y0*y1*y3^2 - 2*y0*y1*y3 - 2*y0*y2*y3 - y0*y2 +
 y0*y3 - y1*y3 + y0 + y3 + 1) *
(-y1 + 1)^-1 * (-y2 + 1)^-1 * (-y0*y2 + 1)^-1 *
(-y1*y3 + 1)^-1 * (-y0*y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 2]: (An equation (1, 0, -1, -1) x - 1 == 0,
         An inequality (-1, 1, 1, 0) x + 1 >= 0,
         An inequality (1, 0, -1, 0) x - 1 >= 0,
         An inequality (0, 0, 1, 0) x + 0 >= 0)
y0 * (-y1 + 1)^-1 * (-y0*y2 + 1)^-1 * (-y0*y1*y3 + 1)^-1
[1, 2]: (An equation (1, -1, -1, 0) x - 1 == 0,
         An inequality (0, -1, 0, 1) x + 0 >= 0,
         An inequality (0, 1, 0, 0) x + 0 >= 0,
         An inequality (1, 0, 0, -1) x + 0 >= 0,
         An inequality (1, -1, 0, 0) x - 1 >= 0)
(-y0^2*y2*y3 + y0*y3 + y0) *
(-y0*y2 + 1)^-1 * (-y0*y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 1, 2]: (An equation (0, 1, 0, -1) x + 0 == 0,
            An equation (1, -1, -1, 0) x - 1 == 0,
            An inequality (0, 1, 0, 0) x + 0 >= 0,
            An inequality (1, -1, 0, 0) x - 1 >= 0)
y0 * (-y0*y2 + 1)^-1 * (-y0*y1*y3 + 1)^-1
[3]: (An inequality (-1, 0, 0, 1) x + 0 >= 0,
      An inequality (0, -1, -1, 1) x - 1 >= 0,
      An inequality (0, 0, 1, 0) x + 0 >= 0,
      An inequality (0, 1, 0, 0) x + 0 >= 0,
      An inequality (1, 0, -1, 0) x + 0 >= 0)
(-y0*y1*y3^2 - y0*y3^2 + y0*y3 + y3) *
(-y3 + 1)^-1 * (-y0*y3 + 1)^-1 *
(-y1*y3 + 1)^-1 * (-y0*y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 3]: (An equation -1 == 0,)
0
[1, 3]: (An equation (1, 0, 0, -1) x + 0 == 0,
         An inequality (1, -1, -1, 0) x - 1 >= 0,
         An inequality (0, 1, 0, 0) x + 0 >= 0,
         An inequality (0, 0, 1, 0) x + 0 >= 0)
y0*y3 * (-y0*y3 + 1)^-1 * (-y0*y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 1, 3]: (An equation -1 == 0,)
0
[2, 3]: (An equation (0, 1, 1, -1) x + 1 == 0,
         An inequality (1, 0, -1, 0) x + 0 >= 0,
         An inequality (-1, 1, 1, 0) x + 1 >= 0,
         An inequality (0, 0, 1, 0) x + 0 >= 0,
         An inequality (0, 1, 0, 0) x + 0 >= 0)
(-y0*y1*y3^2 + y0*y3 + y3) *
(-y1*y3 + 1)^-1 * (-y0*y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 2, 3]: (An equation -1 == 0,)
0
[1, 2, 3]: (An equation (1, 0, 0, -1) x + 0 == 0,
            An equation (1, -1, -1, 0) x - 1 == 0,
            An inequality (0, 1, 0, 0) x + 0 >= 0,
            An inequality (1, -1, 0, 0) x - 1 >= 0)
y0*y3 * (-y0*y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 1, 2, 3]: (An equation -1 == 0,)
0
[4]: (An inequality (-1, -1, 0, 1) x - 1 >= 0,
      An inequality (-1, 0, 1, 0) x + 0 >= 0,
      An inequality (0, 1, 0, 0) x + 0 >= 0,
      An inequality (1, 0, 0, 0) x + 0 >= 0)
y3 * (-y2 + 1)^-1 * (-y3 + 1)^-1 * (-y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 4]: (An equation -1 == 0,)
0
[1, 4]: (An equation -1 == 0,)
0
[0, 1, 4]: (An equation -1 == 0,)
0
[2, 4]: (An equation (1, 1, 0, -1) x + 1 == 0,
         An inequality (-1, 0, 1, 0) x + 0 >= 0,
         An inequality (1, 0, 0, 0) x + 0 >= 0,
         An inequality (0, 1, 0, 0) x + 0 >= 0)
y3 * (-y2 + 1)^-1 * (-y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 2, 4]: (An equation -1 == 0,)
0
[1, 2, 4]: (An equation -1 == 0,)
0
[0, 1, 2, 4]: (An equation -1 == 0,)
0
[3, 4]: (An equation (1, 0, -1, 0) x + 0 == 0,
         An inequality (0, 1, 0, 0) x + 0 >= 0,
         An inequality (-1, -1, 0, 1) x - 1 >= 0,
         An inequality (1, 0, 0, 0) x + 0 >= 0)
y3 * (-y3 + 1)^-1 * (-y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 3, 4]: (An equation -1 == 0,)
0
[1, 3, 4]: (An equation -1 == 0,)
0
[0, 1, 3, 4]: (An equation -1 == 0,)
0
[2, 3, 4]: (An equation (1, 1, 0, -1) x + 1 == 0,
            An equation (1, 0, -1, 0) x + 0 == 0,
            An inequality (0, 1, 0, 0) x + 0 >= 0,
            An inequality (1, 0, 0, 0) x + 0 >= 0)
y3 * (-y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 2, 3, 4]: (An equation -1 == 0,)
0
[1, 2, 3, 4]: (An equation -1 == 0,)
0
[0, 1, 2, 3, 4]: (An equation -1 == 0,)
0

>>> from sage.all import *
>>> P3 = (
...   Polyhedron(
...     ieqs=[(Integer(0), Integer(0), Integer(0), Integer(0), Integer(1)), (Integer(0), Integer(0), Integer(0), Integer(1), Integer(0)),
...           (Integer(0), Integer(0), Integer(1), Integer(0), -Integer(1)), (-Integer(1), Integer(1), Integer(0), -Integer(1), -Integer(1))]),
...   Polyhedron(
...     ieqs=[(Integer(0), Integer(0), -Integer(1), Integer(0), Integer(1)), (Integer(0), Integer(1), Integer(0), Integer(0), -Integer(1)),
...           (Integer(0), Integer(0), Integer(0), Integer(1), Integer(0)), (Integer(0), Integer(0), Integer(1), Integer(0), Integer(0)), (-Integer(1), Integer(1), -Integer(1), -Integer(1), Integer(0))]),
...   Polyhedron(
...     ieqs=[(Integer(1), -Integer(1), Integer(0), Integer(1), Integer(1)), (Integer(1), -Integer(1), Integer(1), Integer(1), Integer(0)),
...           (Integer(0), Integer(0), Integer(0), Integer(0), Integer(1)), (Integer(0), Integer(0), Integer(0), Integer(1), Integer(0)), (Integer(0), Integer(0), Integer(1), Integer(0), Integer(0)),
...           (Integer(1), Integer(0), Integer(1), Integer(1), -Integer(1)), (Integer(0), Integer(1), Integer(0), Integer(0), Integer(0)), (Integer(1), Integer(1), Integer(1), Integer(0), -Integer(1))]),
...   Polyhedron(
...     ieqs=[(Integer(0), Integer(1), Integer(0), -Integer(1), Integer(0)), (Integer(0), -Integer(1), Integer(0), Integer(0), Integer(1)),
...           (-Integer(1), Integer(0), -Integer(1), -Integer(1), Integer(1)), (Integer(0), Integer(0), Integer(1), Integer(0), Integer(0)), (Integer(0), Integer(0), Integer(0), Integer(1), Integer(0))]),
...   Polyhedron(
...     ieqs=[(Integer(0), Integer(1), Integer(0), Integer(0), Integer(0)), (Integer(0), Integer(0), Integer(1), Integer(0), Integer(0)),
...           (-Integer(1), -Integer(1), -Integer(1), Integer(0), Integer(1)), (Integer(0), -Integer(1), Integer(0), Integer(1), Integer(0))]))
>>> def intersect(I):
...     I = iter(I)
...     result = next(I)
...     for i in I:
...         result &= i
...     return result
>>> for J in subsets(range(len(P3))):
...     if not J:
...         continue
...     P = intersect([P3[j] for j in J])
...     print('{}: {}'.format(J, P.Hrepresentation()))
...     print(generating_function_of_integral_points(P, sort_factors=True))
[0]: (An inequality (0, 0, 0, 1) x + 0 >= 0,
      An inequality (0, 0, 1, 0) x + 0 >= 0,
      An inequality (0, 1, 0, -1) x + 0 >= 0,
      An inequality (1, 0, -1, -1) x - 1 >= 0)
y0 * (-y0 + 1)^-1 * (-y1 + 1)^-1 * (-y0*y2 + 1)^-1 * (-y0*y1*y3 + 1)^-1
[1]: (An inequality (0, -1, 0, 1) x + 0 >= 0,
      An inequality (0, 0, 1, 0) x + 0 >= 0,
      An inequality (0, 1, 0, 0) x + 0 >= 0,
      An inequality (1, -1, -1, 0) x - 1 >= 0,
      An inequality (1, 0, 0, -1) x + 0 >= 0)
(-y0^2*y2*y3 - y0^2*y3 + y0*y3 + y0) *
(-y0 + 1)^-1 * (-y0*y2 + 1)^-1 * (-y0*y3 + 1)^-1 *
(-y0*y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 1]: (An equation (0, 1, 0, -1) x + 0 == 0,
         An inequality (1, -1, -1, 0) x - 1 >= 0,
         An inequality (0, 1, 0, 0) x + 0 >= 0,
         An inequality (0, 0, 1, 0) x + 0 >= 0)
y0 * (-y0 + 1)^-1 * (-y0*y2 + 1)^-1 * (-y0*y1*y3 + 1)^-1
[2]: (An inequality (-1, 0, 1, 1) x + 1 >= 0,
     An inequality (-1, 1, 1, 0) x + 1 >= 0,
      An inequality (0, 0, 0, 1) x + 0 >= 0,
      An inequality (0, 0, 1, 0) x + 0 >= 0,
      An inequality (0, 1, 0, 0) x + 0 >= 0,
      An inequality (0, 1, 1, -1) x + 1 >= 0,
      An inequality (1, 0, 0, 0) x + 0 >= 0,
      An inequality (1, 1, 0, -1) x + 1 >= 0)
(y0^2*y1*y2*y3^2 + y0^2*y2^2*y3 + y0*y1^2*y3^2 - y0^2*y2*y3 +
 y0*y1*y2*y3 - y0*y1*y3^2 - 2*y0*y1*y3 - 2*y0*y2*y3 - y0*y2 +
 y0*y3 - y1*y3 + y0 + y3 + 1) *
(-y1 + 1)^-1 * (-y2 + 1)^-1 * (-y0*y2 + 1)^-1 *
(-y1*y3 + 1)^-1 * (-y0*y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 2]: (An equation (1, 0, -1, -1) x - 1 == 0,
         An inequality (-1, 1, 1, 0) x + 1 >= 0,
         An inequality (1, 0, -1, 0) x - 1 >= 0,
         An inequality (0, 0, 1, 0) x + 0 >= 0)
y0 * (-y1 + 1)^-1 * (-y0*y2 + 1)^-1 * (-y0*y1*y3 + 1)^-1
[1, 2]: (An equation (1, -1, -1, 0) x - 1 == 0,
         An inequality (0, -1, 0, 1) x + 0 >= 0,
         An inequality (0, 1, 0, 0) x + 0 >= 0,
         An inequality (1, 0, 0, -1) x + 0 >= 0,
         An inequality (1, -1, 0, 0) x - 1 >= 0)
(-y0^2*y2*y3 + y0*y3 + y0) *
(-y0*y2 + 1)^-1 * (-y0*y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 1, 2]: (An equation (0, 1, 0, -1) x + 0 == 0,
            An equation (1, -1, -1, 0) x - 1 == 0,
            An inequality (0, 1, 0, 0) x + 0 >= 0,
            An inequality (1, -1, 0, 0) x - 1 >= 0)
y0 * (-y0*y2 + 1)^-1 * (-y0*y1*y3 + 1)^-1
[3]: (An inequality (-1, 0, 0, 1) x + 0 >= 0,
      An inequality (0, -1, -1, 1) x - 1 >= 0,
      An inequality (0, 0, 1, 0) x + 0 >= 0,
      An inequality (0, 1, 0, 0) x + 0 >= 0,
      An inequality (1, 0, -1, 0) x + 0 >= 0)
(-y0*y1*y3^2 - y0*y3^2 + y0*y3 + y3) *
(-y3 + 1)^-1 * (-y0*y3 + 1)^-1 *
(-y1*y3 + 1)^-1 * (-y0*y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 3]: (An equation -1 == 0,)
0
[1, 3]: (An equation (1, 0, 0, -1) x + 0 == 0,
         An inequality (1, -1, -1, 0) x - 1 >= 0,
         An inequality (0, 1, 0, 0) x + 0 >= 0,
         An inequality (0, 0, 1, 0) x + 0 >= 0)
y0*y3 * (-y0*y3 + 1)^-1 * (-y0*y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 1, 3]: (An equation -1 == 0,)
0
[2, 3]: (An equation (0, 1, 1, -1) x + 1 == 0,
         An inequality (1, 0, -1, 0) x + 0 >= 0,
         An inequality (-1, 1, 1, 0) x + 1 >= 0,
         An inequality (0, 0, 1, 0) x + 0 >= 0,
         An inequality (0, 1, 0, 0) x + 0 >= 0)
(-y0*y1*y3^2 + y0*y3 + y3) *
(-y1*y3 + 1)^-1 * (-y0*y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 2, 3]: (An equation -1 == 0,)
0
[1, 2, 3]: (An equation (1, 0, 0, -1) x + 0 == 0,
            An equation (1, -1, -1, 0) x - 1 == 0,
            An inequality (0, 1, 0, 0) x + 0 >= 0,
            An inequality (1, -1, 0, 0) x - 1 >= 0)
y0*y3 * (-y0*y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 1, 2, 3]: (An equation -1 == 0,)
0
[4]: (An inequality (-1, -1, 0, 1) x - 1 >= 0,
      An inequality (-1, 0, 1, 0) x + 0 >= 0,
      An inequality (0, 1, 0, 0) x + 0 >= 0,
      An inequality (1, 0, 0, 0) x + 0 >= 0)
y3 * (-y2 + 1)^-1 * (-y3 + 1)^-1 * (-y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 4]: (An equation -1 == 0,)
0
[1, 4]: (An equation -1 == 0,)
0
[0, 1, 4]: (An equation -1 == 0,)
0
[2, 4]: (An equation (1, 1, 0, -1) x + 1 == 0,
         An inequality (-1, 0, 1, 0) x + 0 >= 0,
         An inequality (1, 0, 0, 0) x + 0 >= 0,
         An inequality (0, 1, 0, 0) x + 0 >= 0)
y3 * (-y2 + 1)^-1 * (-y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 2, 4]: (An equation -1 == 0,)
0
[1, 2, 4]: (An equation -1 == 0,)
0
[0, 1, 2, 4]: (An equation -1 == 0,)
0
[3, 4]: (An equation (1, 0, -1, 0) x + 0 == 0,
         An inequality (0, 1, 0, 0) x + 0 >= 0,
         An inequality (-1, -1, 0, 1) x - 1 >= 0,
         An inequality (1, 0, 0, 0) x + 0 >= 0)
y3 * (-y3 + 1)^-1 * (-y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 3, 4]: (An equation -1 == 0,)
0
[1, 3, 4]: (An equation -1 == 0,)
0
[0, 1, 3, 4]: (An equation -1 == 0,)
0
[2, 3, 4]: (An equation (1, 1, 0, -1) x + 1 == 0,
            An equation (1, 0, -1, 0) x + 0 == 0,
            An inequality (0, 1, 0, 0) x + 0 >= 0,
            An inequality (1, 0, 0, 0) x + 0 >= 0)
y3 * (-y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 2, 3, 4]: (An equation -1 == 0,)
0
[1, 2, 3, 4]: (An equation -1 == 0,)
0
[0, 1, 2, 3, 4]: (An equation -1 == 0,)
0

P3 = (
  Polyhedron(
    ieqs=[(0, 0, 0, 0, 1), (0, 0, 0, 1, 0),
          (0, 0, 1, 0, -1), (-1, 1, 0, -1, -1)]),
  Polyhedron(
    ieqs=[(0, 0, -1, 0, 1), (0, 1, 0, 0, -1),
          (0, 0, 0, 1, 0), (0, 0, 1, 0, 0), (-1, 1, -1, -1, 0)]),
  Polyhedron(
    ieqs=[(1, -1, 0, 1, 1), (1, -1, 1, 1, 0),
          (0, 0, 0, 0, 1), (0, 0, 0, 1, 0), (0, 0, 1, 0, 0),
          (1, 0, 1, 1, -1), (0, 1, 0, 0, 0), (1, 1, 1, 0, -1)]),
  Polyhedron(
    ieqs=[(0, 1, 0, -1, 0), (0, -1, 0, 0, 1),
          (-1, 0, -1, -1, 1), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0)]),
  Polyhedron(
    ieqs=[(0, 1, 0, 0, 0), (0, 0, 1, 0, 0),
          (-1, -1, -1, 0, 1), (0, -1, 0, 1, 0)]))
def intersect(I):
    I = iter(I)
    result = next(I)
    for i in I:
        result &= i
    return result
for J in subsets(range(len(P3))):
    if not J:
        continue
    P = intersect([P3[j] for j in J])
    print('{}: {}'.format(J, P.Hrepresentation()))
    print(generating_function_of_integral_points(P, sort_factors=True))
>>> from sage.all import *
>>> P3 = (
...   Polyhedron(
...     ieqs=[(Integer(0), Integer(0), Integer(0), Integer(0), Integer(1)), (Integer(0), Integer(0), Integer(0), Integer(1), Integer(0)),
...           (Integer(0), Integer(0), Integer(1), Integer(0), -Integer(1)), (-Integer(1), Integer(1), Integer(0), -Integer(1), -Integer(1))]),
...   Polyhedron(
...     ieqs=[(Integer(0), Integer(0), -Integer(1), Integer(0), Integer(1)), (Integer(0), Integer(1), Integer(0), Integer(0), -Integer(1)),
...           (Integer(0), Integer(0), Integer(0), Integer(1), Integer(0)), (Integer(0), Integer(0), Integer(1), Integer(0), Integer(0)), (-Integer(1), Integer(1), -Integer(1), -Integer(1), Integer(0))]),
...   Polyhedron(
...     ieqs=[(Integer(1), -Integer(1), Integer(0), Integer(1), Integer(1)), (Integer(1), -Integer(1), Integer(1), Integer(1), Integer(0)),
...           (Integer(0), Integer(0), Integer(0), Integer(0), Integer(1)), (Integer(0), Integer(0), Integer(0), Integer(1), Integer(0)), (Integer(0), Integer(0), Integer(1), Integer(0), Integer(0)),
...           (Integer(1), Integer(0), Integer(1), Integer(1), -Integer(1)), (Integer(0), Integer(1), Integer(0), Integer(0), Integer(0)), (Integer(1), Integer(1), Integer(1), Integer(0), -Integer(1))]),
...   Polyhedron(
...     ieqs=[(Integer(0), Integer(1), Integer(0), -Integer(1), Integer(0)), (Integer(0), -Integer(1), Integer(0), Integer(0), Integer(1)),
...           (-Integer(1), Integer(0), -Integer(1), -Integer(1), Integer(1)), (Integer(0), Integer(0), Integer(1), Integer(0), Integer(0)), (Integer(0), Integer(0), Integer(0), Integer(1), Integer(0))]),
...   Polyhedron(
...     ieqs=[(Integer(0), Integer(1), Integer(0), Integer(0), Integer(0)), (Integer(0), Integer(0), Integer(1), Integer(0), Integer(0)),
...           (-Integer(1), -Integer(1), -Integer(1), Integer(0), Integer(1)), (Integer(0), -Integer(1), Integer(0), Integer(1), Integer(0))]))
>>> def intersect(I):
...     I = iter(I)
...     result = next(I)
...     for i in I:
...         result &= i
...     return result
>>> for J in subsets(range(len(P3))):
...     if not J:
...         continue
...     P = intersect([P3[j] for j in J])
...     print('{}: {}'.format(J, P.Hrepresentation()))
...     print(generating_function_of_integral_points(P, sort_factors=True))
[0]: (An inequality (0, 0, 0, 1) x + 0 >= 0,
      An inequality (0, 0, 1, 0) x + 0 >= 0,
      An inequality (0, 1, 0, -1) x + 0 >= 0,
      An inequality (1, 0, -1, -1) x - 1 >= 0)
y0 * (-y0 + 1)^-1 * (-y1 + 1)^-1 * (-y0*y2 + 1)^-1 * (-y0*y1*y3 + 1)^-1
[1]: (An inequality (0, -1, 0, 1) x + 0 >= 0,
      An inequality (0, 0, 1, 0) x + 0 >= 0,
      An inequality (0, 1, 0, 0) x + 0 >= 0,
      An inequality (1, -1, -1, 0) x - 1 >= 0,
      An inequality (1, 0, 0, -1) x + 0 >= 0)
(-y0^2*y2*y3 - y0^2*y3 + y0*y3 + y0) *
(-y0 + 1)^-1 * (-y0*y2 + 1)^-1 * (-y0*y3 + 1)^-1 *
(-y0*y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 1]: (An equation (0, 1, 0, -1) x + 0 == 0,
         An inequality (1, -1, -1, 0) x - 1 >= 0,
         An inequality (0, 1, 0, 0) x + 0 >= 0,
         An inequality (0, 0, 1, 0) x + 0 >= 0)
y0 * (-y0 + 1)^-1 * (-y0*y2 + 1)^-1 * (-y0*y1*y3 + 1)^-1
[2]: (An inequality (-1, 0, 1, 1) x + 1 >= 0,
     An inequality (-1, 1, 1, 0) x + 1 >= 0,
      An inequality (0, 0, 0, 1) x + 0 >= 0,
      An inequality (0, 0, 1, 0) x + 0 >= 0,
      An inequality (0, 1, 0, 0) x + 0 >= 0,
      An inequality (0, 1, 1, -1) x + 1 >= 0,
      An inequality (1, 0, 0, 0) x + 0 >= 0,
      An inequality (1, 1, 0, -1) x + 1 >= 0)
(y0^2*y1*y2*y3^2 + y0^2*y2^2*y3 + y0*y1^2*y3^2 - y0^2*y2*y3 +
 y0*y1*y2*y3 - y0*y1*y3^2 - 2*y0*y1*y3 - 2*y0*y2*y3 - y0*y2 +
 y0*y3 - y1*y3 + y0 + y3 + 1) *
(-y1 + 1)^-1 * (-y2 + 1)^-1 * (-y0*y2 + 1)^-1 *
(-y1*y3 + 1)^-1 * (-y0*y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 2]: (An equation (1, 0, -1, -1) x - 1 == 0,
         An inequality (-1, 1, 1, 0) x + 1 >= 0,
         An inequality (1, 0, -1, 0) x - 1 >= 0,
         An inequality (0, 0, 1, 0) x + 0 >= 0)
y0 * (-y1 + 1)^-1 * (-y0*y2 + 1)^-1 * (-y0*y1*y3 + 1)^-1
[1, 2]: (An equation (1, -1, -1, 0) x - 1 == 0,
         An inequality (0, -1, 0, 1) x + 0 >= 0,
         An inequality (0, 1, 0, 0) x + 0 >= 0,
         An inequality (1, 0, 0, -1) x + 0 >= 0,
         An inequality (1, -1, 0, 0) x - 1 >= 0)
(-y0^2*y2*y3 + y0*y3 + y0) *
(-y0*y2 + 1)^-1 * (-y0*y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 1, 2]: (An equation (0, 1, 0, -1) x + 0 == 0,
            An equation (1, -1, -1, 0) x - 1 == 0,
            An inequality (0, 1, 0, 0) x + 0 >= 0,
            An inequality (1, -1, 0, 0) x - 1 >= 0)
y0 * (-y0*y2 + 1)^-1 * (-y0*y1*y3 + 1)^-1
[3]: (An inequality (-1, 0, 0, 1) x + 0 >= 0,
      An inequality (0, -1, -1, 1) x - 1 >= 0,
      An inequality (0, 0, 1, 0) x + 0 >= 0,
      An inequality (0, 1, 0, 0) x + 0 >= 0,
      An inequality (1, 0, -1, 0) x + 0 >= 0)
(-y0*y1*y3^2 - y0*y3^2 + y0*y3 + y3) *
(-y3 + 1)^-1 * (-y0*y3 + 1)^-1 *
(-y1*y3 + 1)^-1 * (-y0*y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 3]: (An equation -1 == 0,)
0
[1, 3]: (An equation (1, 0, 0, -1) x + 0 == 0,
         An inequality (1, -1, -1, 0) x - 1 >= 0,
         An inequality (0, 1, 0, 0) x + 0 >= 0,
         An inequality (0, 0, 1, 0) x + 0 >= 0)
y0*y3 * (-y0*y3 + 1)^-1 * (-y0*y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 1, 3]: (An equation -1 == 0,)
0
[2, 3]: (An equation (0, 1, 1, -1) x + 1 == 0,
         An inequality (1, 0, -1, 0) x + 0 >= 0,
         An inequality (-1, 1, 1, 0) x + 1 >= 0,
         An inequality (0, 0, 1, 0) x + 0 >= 0,
         An inequality (0, 1, 0, 0) x + 0 >= 0)
(-y0*y1*y3^2 + y0*y3 + y3) *
(-y1*y3 + 1)^-1 * (-y0*y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 2, 3]: (An equation -1 == 0,)
0
[1, 2, 3]: (An equation (1, 0, 0, -1) x + 0 == 0,
            An equation (1, -1, -1, 0) x - 1 == 0,
            An inequality (0, 1, 0, 0) x + 0 >= 0,
            An inequality (1, -1, 0, 0) x - 1 >= 0)
y0*y3 * (-y0*y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 1, 2, 3]: (An equation -1 == 0,)
0
[4]: (An inequality (-1, -1, 0, 1) x - 1 >= 0,
      An inequality (-1, 0, 1, 0) x + 0 >= 0,
      An inequality (0, 1, 0, 0) x + 0 >= 0,
      An inequality (1, 0, 0, 0) x + 0 >= 0)
y3 * (-y2 + 1)^-1 * (-y3 + 1)^-1 * (-y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 4]: (An equation -1 == 0,)
0
[1, 4]: (An equation -1 == 0,)
0
[0, 1, 4]: (An equation -1 == 0,)
0
[2, 4]: (An equation (1, 1, 0, -1) x + 1 == 0,
         An inequality (-1, 0, 1, 0) x + 0 >= 0,
         An inequality (1, 0, 0, 0) x + 0 >= 0,
         An inequality (0, 1, 0, 0) x + 0 >= 0)
y3 * (-y2 + 1)^-1 * (-y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 2, 4]: (An equation -1 == 0,)
0
[1, 2, 4]: (An equation -1 == 0,)
0
[0, 1, 2, 4]: (An equation -1 == 0,)
0
[3, 4]: (An equation (1, 0, -1, 0) x + 0 == 0,
         An inequality (0, 1, 0, 0) x + 0 >= 0,
         An inequality (-1, -1, 0, 1) x - 1 >= 0,
         An inequality (1, 0, 0, 0) x + 0 >= 0)
y3 * (-y3 + 1)^-1 * (-y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 3, 4]: (An equation -1 == 0,)
0
[1, 3, 4]: (An equation -1 == 0,)
0
[0, 1, 3, 4]: (An equation -1 == 0,)
0
[2, 3, 4]: (An equation (1, 1, 0, -1) x + 1 == 0,
            An equation (1, 0, -1, 0) x + 0 == 0,
            An inequality (0, 1, 0, 0) x + 0 >= 0,
            An inequality (1, 0, 0, 0) x + 0 >= 0)
y3 * (-y1*y3 + 1)^-1 * (-y0*y2*y3 + 1)^-1
[0, 2, 3, 4]: (An equation -1 == 0,)
0
[1, 2, 3, 4]: (An equation -1 == 0,)
0
[0, 1, 2, 3, 4]: (An equation -1 == 0,)
0

P3 = (
  Polyhedron(
    ieqs=[(0, 0, 0, 0, 1), (0, 0, 0, 1, 0),
          (0, 0, 1, 0, -1), (-1, 1, 0, -1, -1)]),
  Polyhedron(
    ieqs=[(0, 0, -1, 0, 1), (0, 1, 0, 0, -1),
          (0, 0, 0, 1, 0), (0, 0, 1, 0, 0), (-1, 1, -1, -1, 0)]),
  Polyhedron(
    ieqs=[(1, -1, 0, 1, 1), (1, -1, 1, 1, 0),
          (0, 0, 0, 0, 1), (0, 0, 0, 1, 0), (0, 0, 1, 0, 0),
          (1, 0, 1, 1, -1), (0, 1, 0, 0, 0), (1, 1, 1, 0, -1)]),
  Polyhedron(
    ieqs=[(0, 1, 0, -1, 0), (0, -1, 0, 0, 1),
          (-1, 0, -1, -1, 1), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0)]),
  Polyhedron(
    ieqs=[(0, 1, 0, 0, 0), (0, 0, 1, 0, 0),
          (-1, -1, -1, 0, 1), (0, -1, 0, 1, 0)]))
def intersect(I):
    I = iter(I)
    result = next(I)
    for i in I:
        result &= i
    return result
for J in subsets(range(len(P3))):
    if not J:
        continue
    P = intersect([P3[j] for j in J])
    print('{}: {}'.format(J, P.Hrepresentation()))
    print(generating_function_of_integral_points(P, sort_factors=True))
sage: P = Polyhedron(vertices=[[1], [5]])
sage: P.generating_function_of_integral_points()
y0^5 + y0^4 + y0^3 + y0^2 + y0

>>> from sage.all import *
>>> P = Polyhedron(vertices=[[Integer(1)], [Integer(5)]])
>>> P.generating_function_of_integral_points()
y0^5 + y0^4 + y0^3 + y0^2 + y0

P = Polyhedron(vertices=[[1], [5]])
P.generating_function_of_integral_points()
>>> from sage.all import *
>>> P = Polyhedron(vertices=[[Integer(1)], [Integer(5)]])
>>> P.generating_function_of_integral_points()
y0^5 + y0^4 + y0^3 + y0^2 + y0

P = Polyhedron(vertices=[[1], [5]])
P.generating_function_of_integral_points()

See also

This function is accessible via sage.geometry.polyhedron.base.Polyhedron_base.generating_function_of_integral_points() as well. More examples can be found there.