Base class for polyhedra over \(\ZZ\)¶

class sage.geometry.polyhedron.base_ZZ.Polyhedron_ZZ(parent, Vrep, Hrep, **kwds)¶

Bases: sage.geometry.polyhedron.base_QQ.Polyhedron_QQ

Base class for Polyhedra over \(\ZZ\)

ehrhart_polynomial(verbose=False, dual=None, irrational_primal=None, irrational_all_primal=None, maxdet=None, no_decomposition=None, compute_vertex_cones=None, smith_form=None, dualization=None, triangulation=None, triangulation_max_height=None, **kwds)¶

Return the Ehrhart polynomial of this polyhedron.

Let \(P\) be a lattice polytope in \(\RR^d\) and define \(L(P,t) = \# (tP \cap \ZZ^d)\). Then E. Ehrhart proved in 1962 that \(L\) coincides with a rational polynomial of degree \(d\) for integer \(t\). \(L\) is called the Ehrhart polynomial of \(P\). For more information see the Wikipedia article Ehrhart_polynomial.

INPUT:

verbose - (boolean, default to False) if True, print the whole output of the LattE command.

The following options are passed to the LattE command, for details you should consult the LattE documentation:

dual - (boolean) triangulate and signed-decompose in the dual space
irrational_primal - (boolean) triangulate in the dual space, signed-decompose in the primal space using irrationalization.
irrational_all_primal - (boolean) Triangulate and signed-decompose in the primal space using irrationalization.
maxdet – (integer) decompose down to an index (determinant) of maxdet instead of index 1 (unimodular cones).
no_decomposition – (boolean) do not signed-decompose simplicial cones.
compute_vertex_cones – (string) either ‘cdd’ or ‘lrs’ or ‘4ti2’
smith_form – (string) either ‘ilio’ or ‘lidia’
dualization – (string) either ‘cdd’ or ‘4ti2’
triangulation - (string) ‘cddlib’, ‘4ti2’ or ‘topcom’
triangulation_max_height - (integer) use a uniform distribution of height from 1 to this number

Note

Any additional argument is forwarded to LattE’s executable count. All occurrences of ‘_’ will be replaced with a ‘-‘.

ALGORITHM:

This method calls the program count from LattE integrale, a program for lattice point enumeration (see https://www.math.ucdavis.edu/~latte/).

See also

latte the interface to LattE integrale

EXAMPLES:

sage: P = Polyhedron(vertices=[(0,0,0),(3,3,3),(-3,2,1),(1,-1,-2)])
sage: p = P.ehrhart_polynomial()    # optional - latte_int
sage: p                             # optional - latte_int
7/2*t^3 + 2*t^2 - 1/2*t + 1
sage: p(1)                          # optional - latte_int
6
sage: len(P.integral_points())
6
sage: p(2)                          # optional - latte_int
36
sage: len((2*P).integral_points())
36

The unit hypercubes:

sage: from itertools import product
sage: def hypercube(d):
....:     return Polyhedron(vertices=list(product([0,1],repeat=d)))
sage: hypercube(3).ehrhart_polynomial()   # optional - latte_int
t^3 + 3*t^2 + 3*t + 1
sage: hypercube(4).ehrhart_polynomial()   # optional - latte_int
t^4 + 4*t^3 + 6*t^2 + 4*t + 1
sage: hypercube(5).ehrhart_polynomial()   # optional - latte_int
t^5 + 5*t^4 + 10*t^3 + 10*t^2 + 5*t + 1
sage: hypercube(6).ehrhart_polynomial()   # optional - latte_int
t^6 + 6*t^5 + 15*t^4 + 20*t^3 + 15*t^2 + 6*t + 1

An empty polyhedron:

sage: P = Polyhedron(ambient_dim=3, vertices=[])
sage: P.ehrhart_polynomial()    # optional - latte_int
0
sage: parent(_)                 # optional - latte_int
Univariate Polynomial Ring in t over Rational Field

fibration_generator(dim)¶

Generate the lattice polytope fibrations.

For the purposes of this function, a lattice polytope fiber is a sub-lattice polytope. Projecting the plane spanned by the subpolytope to a point yields another lattice polytope, the base of the fibration.

INPUT:

dim – integer. The dimension of the lattice polytope fiber.

OUTPUT:

A generator yielding the distinct lattice polytope fibers of given dimension.

EXAMPLES:

sage: P = Polyhedron(toric_varieties.P4_11169().fan().rays(), base_ring=ZZ)
sage: list( P.fibration_generator(2) )
[A 2-dimensional polyhedron in ZZ^4 defined as the convex hull of 3 vertices]

find_translation(translated_polyhedron)¶

Return the translation vector to translated_polyhedron.

INPUT:

translated_polyhedron – a polyhedron.

OUTPUT:

A \(\ZZ\)-vector that translates self to translated_polyhedron. A ValueError is raised if translated_polyhedron is not a translation of self, this can be used to check that two polyhedra are not translates of each other.

EXAMPLES:

sage: X = polytopes.cube()
sage: X.find_translation(X + vector([2,3,5]))
(2, 3, 5)
sage: X.find_translation(2*X)
Traceback (most recent call last):
...
ValueError: polyhedron is not a translation of self

has_IP_property()¶

Test whether the polyhedron has the IP property.

The IP (interior point) property means that

self is compact (a polytope).
self contains the origin as an interior point.

This implies that

self is full-dimensional.
The dual polyhedron is again a polytope (that is, a compact polyhedron), though not necessarily a lattice polytope.

EXAMPLES:

sage: Polyhedron([(1,1),(1,0),(0,1)], base_ring=ZZ).has_IP_property()
False
sage: Polyhedron([(0,0),(1,0),(0,1)], base_ring=ZZ).has_IP_property()
False
sage: Polyhedron([(-1,-1),(1,0),(0,1)], base_ring=ZZ).has_IP_property()
True

REFERENCES:

[PALP]

is_lattice_polytope()¶

Return whether the polyhedron is a lattice polytope.

OUTPUT:

True if the polyhedron is compact and has only integral vertices, False otherwise.

EXAMPLES:

sage: polytopes.cross_polytope(3).is_lattice_polytope()
True
sage: polytopes.regular_polygon(5).is_lattice_polytope()
False

is_reflexive()¶

EXAMPLES:

sage: p = Polyhedron(vertices=[(1,0,0),(0,1,0),(0,0,1),(-1,-1,-1)], base_ring=ZZ)
sage: p.is_reflexive()
True

minkowski_decompositions()¶

Return all Minkowski sums that add up to the polyhedron.

OUTPUT:

A tuple consisting of pairs \((X,Y)\) of \(\ZZ\)-polyhedra that add up to self. All pairs up to exchange of the summands are returned, that is, \((Y,X)\) is not included if \((X,Y)\) already is.

EXAMPLES:

sage: square = Polyhedron(vertices=[(0,0),(1,0),(0,1),(1,1)])
sage: square.minkowski_decompositions()
((A 0-dimensional polyhedron in ZZ^2 defined as the convex hull of 1 vertex,
  A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 4 vertices),
 (A 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices,
  A 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices))

Example from http://cgi.di.uoa.gr/~amantzaf/geo/

 sage: Q = Polyhedron(vertices=[(4,0), (6,0), (0,3), (4,3)])
 sage: R = Polyhedron(vertices=[(0,0), (5,0), (8,4), (3,2)])
 sage: (Q+R).minkowski_decompositions()
 ((A 0-dimensional polyhedron in ZZ^2 defined as the convex hull of 1 vertex,
   A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 7 vertices),
  (A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 4 vertices,
   A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 4 vertices),
  (A 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices,
   A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 7 vertices),
  (A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 5 vertices,
   A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 4 vertices),
  (A 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices,
   A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 7 vertices),
  (A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 5 vertices,
   A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 3 vertices),
  (A 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices,
   A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 7 vertices),
  (A 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices,
   A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 6 vertices))

sage: [ len(square.dilation(i).minkowski_decompositions())
....:   for i in range(6) ]
[1, 2, 5, 8, 13, 18]
sage: [ ceil((i^2+2*i-1)/2)+1 for i in range(10) ]
[1, 2, 5, 8, 13, 18, 25, 32, 41, 50]

polar()¶

Return the polar (dual) polytope.

The polytope must have the IP-property (see has_IP_property()), that is, the origin must be an interior point. In particular, it must be full-dimensional.

OUTPUT:

The polytope whose vertices are the coefficient vectors of the inequalities of self with inhomogeneous term normalized to unity.

EXAMPLES:

sage: p = Polyhedron(vertices=[(1,0,0),(0,1,0),(0,0,1),(-1,-1,-1)], base_ring=ZZ)
sage: p.polar()
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 4 vertices
sage: type(_)
<class 'sage.geometry.polyhedron.parent.Polyhedra_ZZ_ppl_with_category.element_class'>
sage: p.polar().base_ring()
Integer Ring

Base class for polyhedra over \(\ZZ\)¶

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