Base class for polyhedra over \(\ZZ\)¶
- class sage.geometry.polyhedron.base_ZZ.Polyhedron_ZZ(parent, Vrep, Hrep, Vrep_minimal=None, Hrep_minimal=None, pref_rep=None, mutable=False, **kwds)[source]¶
Bases:
Polyhedron_QQBase class for Polyhedra over \(\ZZ\).
- ehrhart_polynomial(engine=None, variable='t', 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)[source]¶
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.
The Ehrhart polynomial may be computed using either LattE Integrale or Normaliz by setting
engineto ‘latte’ or ‘normaliz’ respectively.INPUT:
engine– string; the backend to use. Allowed values are:None(default); When no input is given the Ehrhart polynomial is computed using LattE Integrale (optional)'latte'; use LattE integrale program (optional)'normaliz'; use Normaliz program (optional). The backend ofselfmust be set to ‘normaliz’.
variable– string (default:'t'); the variable in which the Ehrhart polynomial should be expressedWhen the
engineis ‘latte’ or None, the additional input values are:verbose– boolean (default:False); ifTrue, print the whole output of the LattE command.
The following options are passed to the LattE command, for details consult the LattE documentation:
dual– boolean; triangulate and signed-decompose in the dual spaceirrational_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) ofmaxdetinstead 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
OUTPUT:
The Ehrhart polynomial as a univariate polynomial in
variableover a rational field.See also
lattethe interface to LattE Integrale PyNormalizEXAMPLES:
To start, we find the Ehrhart polynomial of a three-dimensional
simplex, first usingengine='latte'. Leaving the engine unspecified sets theengineto ‘latte’ by default:sage: simplex = Polyhedron(vertices=[(0,0,0),(3,3,3),(-3,2,1),(1,-1,-2)]) sage: poly = simplex.ehrhart_polynomial(engine = 'latte') # optional - latte_int sage: poly # optional - latte_int 7/2*t^3 + 2*t^2 - 1/2*t + 1 sage: poly(1) # optional - latte_int 6 sage: len(simplex.integral_points()) 6 sage: poly(2) # optional - latte_int 36 sage: len((2*simplex).integral_points()) 36
>>> from sage.all import * >>> simplex = Polyhedron(vertices=[(Integer(0),Integer(0),Integer(0)),(Integer(3),Integer(3),Integer(3)),(-Integer(3),Integer(2),Integer(1)),(Integer(1),-Integer(1),-Integer(2))]) >>> poly = simplex.ehrhart_polynomial(engine = 'latte') # optional - latte_int >>> poly # optional - latte_int 7/2*t^3 + 2*t^2 - 1/2*t + 1 >>> poly(Integer(1)) # optional - latte_int 6 >>> len(simplex.integral_points()) 6 >>> poly(Integer(2)) # optional - latte_int 36 >>> len((Integer(2)*simplex).integral_points()) 36
Now we find the same Ehrhart polynomial, this time using
engine='normaliz'. To use the Normaliz engine, thesimplexmust be defined withbackend='normaliz':sage: simplex = Polyhedron(vertices=[(0,0,0),(3,3,3),(-3,2,1),(1,-1,-2)], backend='normaliz') # optional - pynormaliz sage: poly = simplex.ehrhart_polynomial(engine='normaliz') # optional - pynormaliz sage: poly # optional - pynormaliz 7/2*t^3 + 2*t^2 - 1/2*t + 1
>>> from sage.all import * >>> simplex = Polyhedron(vertices=[(Integer(0),Integer(0),Integer(0)),(Integer(3),Integer(3),Integer(3)),(-Integer(3),Integer(2),Integer(1)),(Integer(1),-Integer(1),-Integer(2))], backend='normaliz') # optional - pynormaliz >>> poly = simplex.ehrhart_polynomial(engine='normaliz') # optional - pynormaliz >>> poly # optional - pynormaliz 7/2*t^3 + 2*t^2 - 1/2*t + 1
If the
engine='normaliz', the backend should be'normaliz', otherwise it returns an error:sage: simplex = Polyhedron(vertices=[(0,0,0),(3,3,3),(-3,2,1),(1,-1,-2)]) sage: simplex.ehrhart_polynomial(engine='normaliz') Traceback (most recent call last): ... TypeError: The polyhedron's backend should be 'normaliz'
>>> from sage.all import * >>> simplex = Polyhedron(vertices=[(Integer(0),Integer(0),Integer(0)),(Integer(3),Integer(3),Integer(3)),(-Integer(3),Integer(2),Integer(1)),(Integer(1),-Integer(1),-Integer(2))]) >>> simplex.ehrhart_polynomial(engine='normaliz') Traceback (most recent call last): ... TypeError: The polyhedron's backend should be 'normaliz'
Now we find the Ehrhart polynomials of the unit hypercubes of dimensions three through six. They are computed first with
engine='latte'and then withengine='normaliz'. The degree of the Ehrhart polynomial matches the dimension of the hypercube, and the coefficient of the leading monomial equals the volume of the unit hypercube:sage: # optional - latte_int sage: from itertools import product sage: def hypercube(d): ....: return Polyhedron(vertices=list(product([0,1],repeat=d))) sage: hypercube(3).ehrhart_polynomial() t^3 + 3*t^2 + 3*t + 1 sage: hypercube(4).ehrhart_polynomial() t^4 + 4*t^3 + 6*t^2 + 4*t + 1 sage: hypercube(5).ehrhart_polynomial() t^5 + 5*t^4 + 10*t^3 + 10*t^2 + 5*t + 1 sage: hypercube(6).ehrhart_polynomial() t^6 + 6*t^5 + 15*t^4 + 20*t^3 + 15*t^2 + 6*t + 1 sage: # optional - pynormaliz sage: from itertools import product sage: def hypercube(d): ....: return Polyhedron(vertices=list(product([0,1],repeat=d)),backend='normaliz') sage: hypercube(3).ehrhart_polynomial(engine='normaliz') t^3 + 3*t^2 + 3*t + 1 sage: hypercube(4).ehrhart_polynomial(engine='normaliz') t^4 + 4*t^3 + 6*t^2 + 4*t + 1 sage: hypercube(5).ehrhart_polynomial(engine='normaliz') t^5 + 5*t^4 + 10*t^3 + 10*t^2 + 5*t + 1 sage: hypercube(6).ehrhart_polynomial(engine='normaliz') t^6 + 6*t^5 + 15*t^4 + 20*t^3 + 15*t^2 + 6*t + 1
>>> from sage.all import * >>> # optional - latte_int >>> from itertools import product >>> def hypercube(d): ... return Polyhedron(vertices=list(product([Integer(0),Integer(1)],repeat=d))) >>> hypercube(Integer(3)).ehrhart_polynomial() t^3 + 3*t^2 + 3*t + 1 >>> hypercube(Integer(4)).ehrhart_polynomial() t^4 + 4*t^3 + 6*t^2 + 4*t + 1 >>> hypercube(Integer(5)).ehrhart_polynomial() t^5 + 5*t^4 + 10*t^3 + 10*t^2 + 5*t + 1 >>> hypercube(Integer(6)).ehrhart_polynomial() t^6 + 6*t^5 + 15*t^4 + 20*t^3 + 15*t^2 + 6*t + 1 >>> # optional - pynormaliz >>> from itertools import product >>> def hypercube(d): ... return Polyhedron(vertices=list(product([Integer(0),Integer(1)],repeat=d)),backend='normaliz') >>> hypercube(Integer(3)).ehrhart_polynomial(engine='normaliz') t^3 + 3*t^2 + 3*t + 1 >>> hypercube(Integer(4)).ehrhart_polynomial(engine='normaliz') t^4 + 4*t^3 + 6*t^2 + 4*t + 1 >>> hypercube(Integer(5)).ehrhart_polynomial(engine='normaliz') t^5 + 5*t^4 + 10*t^3 + 10*t^2 + 5*t + 1 >>> hypercube(Integer(6)).ehrhart_polynomial(engine='normaliz') 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() 0 sage: parent(_) Univariate Polynomial Ring in t over Rational Field
>>> from sage.all import * >>> p = Polyhedron(ambient_dim=Integer(3), vertices=[]) >>> p.ehrhart_polynomial() 0 >>> parent(_) Univariate Polynomial Ring in t over Rational Field
The polyhedron should be compact:
sage: C = Polyhedron(rays=[[1,2],[2,1]]) sage: C.ehrhart_polynomial() Traceback (most recent call last): ... ValueError: Ehrhart polynomial only defined for compact polyhedra
>>> from sage.all import * >>> C = Polyhedron(rays=[[Integer(1),Integer(2)],[Integer(2),Integer(1)]]) >>> C.ehrhart_polynomial() Traceback (most recent call last): ... ValueError: Ehrhart polynomial only defined for compact polyhedra
- fibration_generator(dim)[source]¶
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) # needs palp sage.graphs sage: list(P.fibration_generator(2)) # needs palp sage.graphs [A 2-dimensional polyhedron in ZZ^4 defined as the convex hull of 3 vertices]
>>> from sage.all import * >>> P = Polyhedron(toric_varieties.P4_11169().fan().rays(), base_ring=ZZ) # needs palp sage.graphs >>> list(P.fibration_generator(Integer(2))) # needs palp sage.graphs [A 2-dimensional polyhedron in ZZ^4 defined as the convex hull of 3 vertices]
- find_translation(translated_polyhedron)[source]¶
Return the translation vector to
translated_polyhedron.INPUT:
translated_polyhedron– a polyhedron
OUTPUT:
A \(\ZZ\)-vector that translates
selftotranslated_polyhedron. AValueErroris raised iftranslated_polyhedronis not a translation ofself, 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
>>> from sage.all import * >>> X = polytopes.cube() >>> X.find_translation(X + vector([Integer(2),Integer(3),Integer(5)])) (2, 3, 5) >>> X.find_translation(Integer(2)*X) Traceback (most recent call last): ... ValueError: polyhedron is not a translation of self
- has_IP_property()[source]¶
Test whether the polyhedron has the IP property.
The IP (interior point) property means that
selfis compact (a polytope).selfcontains the origin as an interior point.
This implies that
selfis 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
>>> from sage.all import * >>> Polyhedron([(Integer(1),Integer(1)),(Integer(1),Integer(0)),(Integer(0),Integer(1))], base_ring=ZZ).has_IP_property() False >>> Polyhedron([(Integer(0),Integer(0)),(Integer(1),Integer(0)),(Integer(0),Integer(1))], base_ring=ZZ).has_IP_property() False >>> Polyhedron([(-Integer(1),-Integer(1)),(Integer(1),Integer(0)),(Integer(0),Integer(1))], base_ring=ZZ).has_IP_property() True
REFERENCES:
- is_lattice_polytope()[source]¶
Return whether the polyhedron is a lattice polytope.
OUTPUT:
Trueif the polyhedron is compact and has only integral vertices,Falseotherwise.EXAMPLES:
sage: polytopes.cross_polytope(3).is_lattice_polytope() True sage: polytopes.regular_polygon(5).is_lattice_polytope() # needs sage.rings.number_field False
>>> from sage.all import * >>> polytopes.cross_polytope(Integer(3)).is_lattice_polytope() True >>> polytopes.regular_polygon(Integer(5)).is_lattice_polytope() # needs sage.rings.number_field False
- is_reflexive()[source]¶
A lattice polytope is reflexive if it contains the origin in its interior and its polar with respect to the origin is a lattice polytope.
Equivalently, it is reflexive if it is of the form \(\{x \in \mathbb{R}^d: Ax \leq 1\}\) for some integer matrix \(A\) and \(d\) the ambient dimension.
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 sage: polytopes.hypercube(4).is_reflexive() True sage: p = Polyhedron(vertices=[(1,0), (0,2), (-1,0), (0,-1)], base_ring=ZZ) sage: p.is_reflexive() False sage: p = Polyhedron(vertices=[(1,0), (0,2), (-1,0)], base_ring=ZZ) sage: p.is_reflexive() False
>>> from sage.all import * >>> p = Polyhedron(vertices=[(Integer(1),Integer(0),Integer(0)),(Integer(0),Integer(1),Integer(0)),(Integer(0),Integer(0),Integer(1)),(-Integer(1),-Integer(1),-Integer(1))], base_ring=ZZ) >>> p.is_reflexive() True >>> polytopes.hypercube(Integer(4)).is_reflexive() True >>> p = Polyhedron(vertices=[(Integer(1),Integer(0)), (Integer(0),Integer(2)), (-Integer(1),Integer(0)), (Integer(0),-Integer(1))], base_ring=ZZ) >>> p.is_reflexive() False >>> p = Polyhedron(vertices=[(Integer(1),Integer(0)), (Integer(0),Integer(2)), (-Integer(1),Integer(0))], base_ring=ZZ) >>> p.is_reflexive() False
An error is raised, if the polyhedron is not compact:
sage: p = Polyhedron(rays=[(1,)], base_ring=ZZ) sage: p.is_reflexive() Traceback (most recent call last): ... ValueError: the polyhedron is not compact
>>> from sage.all import * >>> p = Polyhedron(rays=[(Integer(1),)], base_ring=ZZ) >>> p.is_reflexive() Traceback (most recent call last): ... ValueError: the polyhedron is not compact
- minkowski_decompositions()[source]¶
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))
>>> from sage.all import * >>> square = Polyhedron(vertices=[(Integer(0),Integer(0)),(Integer(1),Integer(0)),(Integer(0),Integer(1)),(Integer(1),Integer(1))]) >>> 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: [ integer_ceil((i^2 + 2*i - 1) / 2) + 1 for i in range(10) ] [1, 2, 5, 8, 13, 18, 25, 32, 41, 50]
- normal_form(algorithm='palp_native', permutation=False)[source]¶
Return the normal form of vertices of the lattice polytope
self.INPUT:
algorithm– must be'palp_native', the defaultpermutation– boolean (default:False); ifTrue, the permutation applied to vertices to obtain the normal form is returned as well
For more more detail, see
normal_form().EXAMPLES:
We compute the normal form of the “diamond”:
sage: d = Polyhedron([(1,0), (0,1), (-1,0), (0,-1)]) sage: d.vertices() (A vertex at (-1, 0), A vertex at (0, -1), A vertex at (0, 1), A vertex at (1, 0)) sage: d.normal_form() # needs sage.groups [(1, 0), (0, 1), (0, -1), (-1, 0)] sage: d.lattice_polytope().normal_form("palp_native") # needs sage.groups M( 1, 0), M( 0, 1), M( 0, -1), M(-1, 0) in 2-d lattice M
>>> from sage.all import * >>> d = Polyhedron([(Integer(1),Integer(0)), (Integer(0),Integer(1)), (-Integer(1),Integer(0)), (Integer(0),-Integer(1))]) >>> d.vertices() (A vertex at (-1, 0), A vertex at (0, -1), A vertex at (0, 1), A vertex at (1, 0)) >>> d.normal_form() # needs sage.groups [(1, 0), (0, 1), (0, -1), (-1, 0)] >>> d.lattice_polytope().normal_form("palp_native") # needs sage.groups M( 1, 0), M( 0, 1), M( 0, -1), M(-1, 0) in 2-d lattice M
Using
permutation=True:sage: d.normal_form(permutation=True) # needs sage.groups ([(1, 0), (0, 1), (0, -1), (-1, 0)], ())
>>> from sage.all import * >>> d.normal_form(permutation=True) # needs sage.groups ([(1, 0), (0, 1), (0, -1), (-1, 0)], ())
It is not possible to compute normal forms for polytopes which do not span the space:
sage: p = Polyhedron([(1,0,0), (0,1,0), (-1,0,0), (0,-1,0)]) sage: p.normal_form() Traceback (most recent call last): ... ValueError: normal form is not defined for lower-dimensional polyhedra, got A 2-dimensional polyhedron in ZZ^3 defined as the convex hull of 4 vertices
>>> from sage.all import * >>> p = Polyhedron([(Integer(1),Integer(0),Integer(0)), (Integer(0),Integer(1),Integer(0)), (-Integer(1),Integer(0),Integer(0)), (Integer(0),-Integer(1),Integer(0))]) >>> p.normal_form() Traceback (most recent call last): ... ValueError: normal form is not defined for lower-dimensional polyhedra, got A 2-dimensional polyhedron in ZZ^3 defined as the convex hull of 4 vertices
The normal form is also not defined for unbounded polyhedra:
sage: p = Polyhedron(vertices=[[1, 1]], rays=[[1, 0], [0, 1]], base_ring=ZZ) sage: p.normal_form() Traceback (most recent call last): ... ValueError: normal form is not defined for unbounded polyhedra, got A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 1 vertex and 2 rays
>>> from sage.all import * >>> p = Polyhedron(vertices=[[Integer(1), Integer(1)]], rays=[[Integer(1), Integer(0)], [Integer(0), Integer(1)]], base_ring=ZZ) >>> p.normal_form() Traceback (most recent call last): ... ValueError: normal form is not defined for unbounded polyhedra, got A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 1 vertex and 2 rays
See Issue #15280 for proposed extensions to these cases.
- polar()[source]¶
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
selfwith 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
>>> from sage.all import * >>> p = Polyhedron(vertices=[(Integer(1),Integer(0),Integer(0)),(Integer(0),Integer(1),Integer(0)),(Integer(0),Integer(0),Integer(1)),(-Integer(1),-Integer(1),-Integer(1))], base_ring=ZZ) >>> p.polar() A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 4 vertices >>> type(_) <class 'sage.geometry.polyhedron.parent.Polyhedra_ZZ_ppl_with_category.element_class'> >>> p.polar().base_ring() Integer Ring