Hyperplane Arrangements¶

Before talking about hyperplane arrangements, let us start with individual hyperplanes. This package uses certain linear expressions to represent hyperplanes, that is, a linear expression \(3x + 3y - 5z - 7\) stands for the hyperplane with the equation \(3x + 3y - 5z = 7\). To create it in Sage, you first have to create a HyperplaneArrangements object to define the variables \(x\), \(y\), \(z\):

Sage

sage: H.<x,y,z> = HyperplaneArrangements(QQ)
sage: h = 3*x + 2*y - 5*z - 7;  h
Hyperplane 3*x + 2*y - 5*z - 7
sage: h.normal()
(3, 2, -5)
sage: h.constant_term()
-7

Python

>>> from sage.all import *
>>> H = HyperplaneArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = H._first_ngens(3)
>>> h = Integer(3)*x + Integer(2)*y - Integer(5)*z - Integer(7);  h
Hyperplane 3*x + 2*y - 5*z - 7
>>> h.normal()
(3, 2, -5)
>>> h.constant_term()
-7

Sage Live

H.<x,y,z> = HyperplaneArrangements(QQ)
h = 3*x + 2*y - 5*z - 7;  h
h.normal()
h.constant_term()

The individual hyperplanes behave like the linear expression with regard to addition and scalar multiplication, which is why you can do linear combinations of the coordinates:

Sage

sage: -2*h
Hyperplane -6*x - 4*y + 10*z + 14
sage: x, y, z
(Hyperplane x + 0*y + 0*z + 0,
 Hyperplane 0*x + y + 0*z + 0,
 Hyperplane 0*x + 0*y + z + 0)

Python

>>> from sage.all import *
>>> -Integer(2)*h
Hyperplane -6*x - 4*y + 10*z + 14
>>> x, y, z
(Hyperplane x + 0*y + 0*z + 0,
 Hyperplane 0*x + y + 0*z + 0,
 Hyperplane 0*x + 0*y + z + 0)

Sage Live

-2*h
x, y, z

See sage.geometry.hyperplane_arrangement.hyperplane for more functionality of the individual hyperplanes.

class sage.geometry.hyperplane_arrangement.arrangement.HyperplaneArrangementElement(parent, hyperplanes, check=True, backend=None)[source]¶

Bases: Element

A hyperplane arrangement.

Warning

You should never create HyperplaneArrangementElement instances directly, always use the parent.

add_hyperplane(other)[source]¶

The union of self with other.

INPUT:

other – a hyperplane arrangement or something that can be converted into a hyperplane arrangement

OUTPUT: a new hyperplane arrangement

EXAMPLES:

Sage

sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: A = H([1,2,3], [0,1,1], [0,1,-1], [1,-1,0], [1,1,0])
sage: B = H([1,1,1], [1,-1,1], [1,0,-1])
sage: C = A.union(B); C
Arrangement of 8 hyperplanes of dimension 2 and rank 2
sage: C == A | B   # syntactic sugar
True

Python

>>> from sage.all import *
>>> H = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H([Integer(1),Integer(2),Integer(3)], [Integer(0),Integer(1),Integer(1)], [Integer(0),Integer(1),-Integer(1)], [Integer(1),-Integer(1),Integer(0)], [Integer(1),Integer(1),Integer(0)])
>>> B = H([Integer(1),Integer(1),Integer(1)], [Integer(1),-Integer(1),Integer(1)], [Integer(1),Integer(0),-Integer(1)])
>>> C = A.union(B); C
Arrangement of 8 hyperplanes of dimension 2 and rank 2
>>> C == A | B   # syntactic sugar
True

Sage Live

H.<x,y> = HyperplaneArrangements(QQ)
A = H([1,2,3], [0,1,1], [0,1,-1], [1,-1,0], [1,1,0])
B = H([1,1,1], [1,-1,1], [1,0,-1])
C = A.union(B); C
C == A | B   # syntactic sugar

A single hyperplane is coerced into a hyperplane arrangement if necessary:

Sage

sage: A.union(x+y-1)
Arrangement of 6 hyperplanes of dimension 2 and rank 2
sage: A.add_hyperplane(x+y-1)    # alias
Arrangement of 6 hyperplanes of dimension 2 and rank 2

sage: P.<x,y> = HyperplaneArrangements(RR)
sage: C = P(2*x + 4*y + 5)
sage: C.union(A)
Arrangement of 6 hyperplanes of dimension 2 and rank 2

Python

>>> from sage.all import *
>>> A.union(x+y-Integer(1))
Arrangement of 6 hyperplanes of dimension 2 and rank 2
>>> A.add_hyperplane(x+y-Integer(1))    # alias
Arrangement of 6 hyperplanes of dimension 2 and rank 2

>>> P = HyperplaneArrangements(RR, names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> C = P(Integer(2)*x + Integer(4)*y + Integer(5))
>>> C.union(A)
Arrangement of 6 hyperplanes of dimension 2 and rank 2

Sage Live

A.union(x+y-1)
A.add_hyperplane(x+y-1)    # alias
P.<x,y> = HyperplaneArrangements(RR)
C = P(2*x + 4*y + 5)
C.union(A)

backend()[source]¶

Return the backend used for polyhedral objects.

OUTPUT: string giving the backend or None if none is specified

EXAMPLES:

By default, no backend is specified:

Sage

sage: H = HyperplaneArrangements(QQ)
sage: A = H()
sage: A.backend()

Python

>>> from sage.all import *
>>> H = HyperplaneArrangements(QQ)
>>> A = H()
>>> A.backend()

Sage Live

H = HyperplaneArrangements(QQ)
A = H()
A.backend()

Otherwise, one may specify a polyhedral backend:

Sage

sage: A = H(backend='ppl')
sage: A.backend()
'ppl'
sage: A = H(backend='normaliz')
sage: A.backend()
'normaliz'

Python

>>> from sage.all import *
>>> A = H(backend='ppl')
>>> A.backend()
'ppl'
>>> A = H(backend='normaliz')
>>> A.backend()
'normaliz'

Sage Live

A = H(backend='ppl')
A.backend()
A = H(backend='normaliz')
A.backend()

bounded_regions()[source]¶

Return the relatively bounded regions of the arrangement.

A region is relatively bounded if its intersection with the space spanned by the normals to the hyperplanes is bounded. This is the same as being bounded in the case that the hyperplane arrangement is essential. It is assumed that the arrangement is defined over the rationals.

OUTPUT:

Tuple of polyhedra. The relatively bounded regions of the arrangement.

See also

unbounded_regions()

EXAMPLES:

Sage

sage: # needs sage.combinat
sage: A = hyperplane_arrangements.semiorder(3)
sage: A.bounded_regions()
(A 3-dimensional polyhedron in QQ^3 defined
     as the convex hull of 3 vertices and 1 line,
 A 3-dimensional polyhedron in QQ^3 defined
     as the convex hull of 3 vertices and 1 line,
 A 3-dimensional polyhedron in QQ^3 defined
     as the convex hull of 3 vertices and 1 line,
 A 3-dimensional polyhedron in QQ^3 defined
     as the convex hull of 6 vertices and 1 line,
 A 3-dimensional polyhedron in QQ^3 defined
     as the convex hull of 3 vertices and 1 line,
 A 3-dimensional polyhedron in QQ^3 defined
     as the convex hull of 3 vertices and 1 line,
 A 3-dimensional polyhedron in QQ^3 defined
     as the convex hull of 3 vertices and 1 line)
sage: A.bounded_regions()[0].is_compact()    # the regions are only *relatively* bounded
False
sage: A.is_essential()
False

Python

>>> from sage.all import *
>>> # needs sage.combinat
>>> A = hyperplane_arrangements.semiorder(Integer(3))
>>> A.bounded_regions()
(A 3-dimensional polyhedron in QQ^3 defined
     as the convex hull of 3 vertices and 1 line,
 A 3-dimensional polyhedron in QQ^3 defined
     as the convex hull of 3 vertices and 1 line,
 A 3-dimensional polyhedron in QQ^3 defined
     as the convex hull of 3 vertices and 1 line,
 A 3-dimensional polyhedron in QQ^3 defined
     as the convex hull of 6 vertices and 1 line,
 A 3-dimensional polyhedron in QQ^3 defined
     as the convex hull of 3 vertices and 1 line,
 A 3-dimensional polyhedron in QQ^3 defined
     as the convex hull of 3 vertices and 1 line,
 A 3-dimensional polyhedron in QQ^3 defined
     as the convex hull of 3 vertices and 1 line)
>>> A.bounded_regions()[Integer(0)].is_compact()    # the regions are only *relatively* bounded
False
>>> A.is_essential()
False

Sage Live

# needs sage.combinat
A = hyperplane_arrangements.semiorder(3)
A.bounded_regions()
A.bounded_regions()[0].is_compact()    # the regions are only *relatively* bounded
A.is_essential()

center()[source]¶

Return the center of the hyperplane arrangement.

The polyhedron defined to be the set of all points in the ambient space of the arrangement that lie on all of the hyperplanes.

OUTPUT: a polyhedron

EXAMPLES:

The empty hyperplane arrangement has the entire ambient space as its center:

Sage

sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: A = H()
sage: A.center()
A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 1 vertex and 2 lines

Python

>>> from sage.all import *
>>> H = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H()
>>> A.center()
A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 1 vertex and 2 lines

Sage Live

H.<x,y> = HyperplaneArrangements(QQ)
A = H()
A.center()

The Shi arrangement in dimension 3 has an empty center:

Sage

sage: A = hyperplane_arrangements.Shi(3)
sage: A.center()
The empty polyhedron in QQ^3

Python

>>> from sage.all import *
>>> A = hyperplane_arrangements.Shi(Integer(3))
>>> A.center()
The empty polyhedron in QQ^3

Sage Live

A = hyperplane_arrangements.Shi(3)
A.center()

The Braid arrangement in dimension 3 has a center that is neither empty nor full-dimensional:

Sage

sage: A = hyperplane_arrangements.braid(3)                                  # needs sage.combinat
sage: A.center()                                                            # needs sage.combinat
A 1-dimensional polyhedron in QQ^3 defined as the convex hull of 1 vertex and 1 line

Python

>>> from sage.all import *
>>> A = hyperplane_arrangements.braid(Integer(3))                                  # needs sage.combinat
>>> A.center()                                                            # needs sage.combinat
A 1-dimensional polyhedron in QQ^3 defined as the convex hull of 1 vertex and 1 line

Sage Live

A = hyperplane_arrangements.braid(3)                                  # needs sage.combinat
A.center()                                                            # needs sage.combinat

change_ring(base_ring)[source]¶

Return hyperplane arrangement over the new base ring.

INPUT:

base_ring – the new base ring; must be a field for hyperplane arrangements

OUTPUT:

The hyperplane arrangement obtained by changing the base field, as a new hyperplane arrangement.

Warning

While there is often a one-to-one correspondence between the hyperplanes of self and those of self.change_ring(base_ring), there is no guarantee that the order in which they appear in self.hyperplanes() will match the order in which their counterparts in self.cone() will appear in self.change_ring(base_ring).hyperplanes()!

EXAMPLES:

Sage

sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: A = H([(1,1), 0], [(2,3), -1])
sage: A.change_ring(FiniteField(2))
Arrangement <y + 1 | x + y>

Python

>>> from sage.all import *
>>> H = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H([(Integer(1),Integer(1)), Integer(0)], [(Integer(2),Integer(3)), -Integer(1)])
>>> A.change_ring(FiniteField(Integer(2)))
Arrangement <y + 1 | x + y>

Sage Live

H.<x,y> = HyperplaneArrangements(QQ)
A = H([(1,1), 0], [(2,3), -1])
A.change_ring(FiniteField(2))

characteristic_polynomial()[source]¶

Return the characteristic polynomial of the hyperplane arrangement.

OUTPUT: the characteristic polynomial in \(\QQ[x]\)

EXAMPLES:

Sage

sage: a = hyperplane_arrangements.coordinate(2)
sage: a.characteristic_polynomial()
x^2 - 2*x + 1

Python

>>> from sage.all import *
>>> a = hyperplane_arrangements.coordinate(Integer(2))
>>> a.characteristic_polynomial()
x^2 - 2*x + 1

Sage Live

a = hyperplane_arrangements.coordinate(2)
a.characteristic_polynomial()

closed_faces(labelled=True)[source]¶

Return the closed faces of the hyperplane arrangement self (provided that self is defined over a totally ordered field).

Let \(\mathcal{A}\) be a hyperplane arrangement in the vector space \(K^n\), whose hyperplanes are the zero sets of the affine-linear functions \(u_1, u_2, \ldots, u_N\). (We consider these functions \(u_1, u_2, \ldots, u_N\), and not just the hyperplanes, as given. We also assume the field \(K\) to be totally ordered.) For any point \(x \in K^n\), we define the sign vector of \(x\) to be the vector \((v_1, v_2, \ldots, v_N) \in \{-1, 0, 1\}^N\) such that (for each \(i\)) the number \(v_i\) is the sign of \(u_i(x)\). For any \(v \in \{-1, 0, 1\}^N\), we let \(F_v\) be the set of all \(x \in K^n\) which have sign vector \(v\). The nonempty ones among all these subsets \(F_v\) are called the open faces of \(\mathcal{A}\). They form a partition of the set \(K^n\).

Furthermore, for any \(v = (v_1, v_2, \ldots, v_N) \in \{-1, 0, 1\}^N\), we let \(G_v\) be the set of all \(x \in K^n\) such that, for every \(i\), the sign of \(u_i(x)\) is either \(0\) or \(v_i\). Then, \(G_v\) is a polyhedron. The nonempty ones among all these polyhedra \(G_v\) are called the closed faces of \(\mathcal{A}\). While several sign vectors \(v\) can lead to one and the same closed face \(G_v\), we can assign to every closed face a canonical choice of a sign vector: Namely, if \(G\) is a closed face of \(\mathcal{A}\), then the sign vector of \(G\) is defined to be the vector \((v_1, v_2, \ldots, v_N) \in \{-1, 0, 1\}^N\) where \(x\) is any point in the relative interior of \(G\) and where, for each \(i\), the number \(v_i\) is the sign of \(u_i(x)\). (This does not depend on the choice of \(x\).)

There is a one-to-one correspondence between the closed faces and the open faces of \(\mathcal{A}\). It sends a closed face \(G\) to the open face \(F_v\), where \(v\) is the sign vector of \(G\); this \(F_v\) is also the relative interior of \(G_v\). The inverse map sends any open face \(O\) to the closure of \(O\).

INPUT:

labelled – boolean (default: True); if True, then this method returns not the faces itself but rather pairs \((v, F)\) where \(F\) is a closed face and \(v\) is its sign vector (here, the order and the orientation of the \(u_1, u_2, \ldots, u_N\) is as given by self.hyperplanes()).

OUTPUT:

A tuple containing the closed faces as polyhedra, or (if labelled is set to True) the pairs of sign vectors and corresponding closed faces.

Todo

Should the output rather be a dictionary where the keys are the sign vectors and the values are the faces?

EXAMPLES:

Sage

sage: # needs sage.graphs
sage: a = hyperplane_arrangements.braid(2)
sage: a.hyperplanes()
(Hyperplane t0 - t1 + 0,)
sage: a.closed_faces()
(((0,),  A 1-dimensional polyhedron in QQ^2 defined
         as the convex hull of 1 vertex and 1 line),
 ((1,),  A 2-dimensional polyhedron in QQ^2 defined
         as the convex hull of 1 vertex, 1 ray, 1 line),
 ((-1,), A 2-dimensional polyhedron in QQ^2 defined
         as the convex hull of 1 vertex, 1 ray, 1 line))
sage: a.closed_faces(labelled=False)
(A 1-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex and 1 line,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex, 1 ray, 1 line,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex, 1 ray, 1 line)
sage: [(v, F, F.representative_point()) for v, F in a.closed_faces()]
[((0,),  A 1-dimensional polyhedron in QQ^2 defined
         as the convex hull of 1 vertex and 1 line,      (0, 0)),
 ((1,),  A 2-dimensional polyhedron in QQ^2 defined
         as the convex hull of 1 vertex, 1 ray, 1 line,  (0, -1)),
 ((-1,), A 2-dimensional polyhedron in QQ^2 defined
         as the convex hull of 1 vertex, 1 ray, 1 line,  (-1, 0))]

sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: a = H(x, y+1)
sage: a.hyperplanes()
(Hyperplane 0*x + y + 1, Hyperplane x + 0*y + 0)
sage: [(v, F, F.representative_point()) for v, F in a.closed_faces()]
[((0, 0),   A 0-dimensional polyhedron in QQ^2 defined
            as the convex hull of 1 vertex,             (0, -1)),
 ((0, 1),   A 1-dimensional polyhedron in QQ^2 defined
            as the convex hull of 1 vertex and 1 ray,   (1, -1)),
 ((0, -1),  A 1-dimensional polyhedron in QQ^2 defined
            as the convex hull of 1 vertex and 1 ray,   (-1, -1)),
 ((1, 0),   A 1-dimensional polyhedron in QQ^2 defined
            as the convex hull of 1 vertex and 1 ray,   (0, 0)),
 ((1, 1),   A 2-dimensional polyhedron in QQ^2 defined
            as the convex hull of 1 vertex and 2 rays,  (1, 0)),
 ((1, -1),  A 2-dimensional polyhedron in QQ^2 defined
            as the convex hull of 1 vertex and 2 rays,  (-1, 0)),
 ((-1, 0),  A 1-dimensional polyhedron in QQ^2 defined
            as the convex hull of 1 vertex and 1 ray,   (0, -2)),
 ((-1, 1),  A 2-dimensional polyhedron in QQ^2 defined
            as the convex hull of 1 vertex and 2 rays,  (1, -2)),
 ((-1, -1), A 2-dimensional polyhedron in QQ^2 defined
            as the convex hull of 1 vertex and 2 rays,  (-1, -2))]

sage: # needs sage.graphs
sage: a = hyperplane_arrangements.braid(3)
sage: a.hyperplanes()
(Hyperplane 0*t0 + t1 - t2 + 0,
 Hyperplane t0 - t1 + 0*t2 + 0,
 Hyperplane t0 + 0*t1 - t2 + 0)
sage: [(v, F, F.representative_point()) for v, F in a.closed_faces()]
[((0, 0, 0),    A 1-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex and 1 line,      (0, 0, 0)),
 ((0, 1, 1),    A 2-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 1 ray, 1 line,  (0, -1, -1)),
 ((0, -1, -1),  A 2-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 1 ray, 1 line,  (-1, 0, 0)),
 ((1, 0, 1),    A 2-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 1 ray, 1 line,  (1, 1, 0)),
 ((1, 1, 1),    A 3-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 2 rays, 1 line, (0, -1, -2)),
 ((1, -1, 0),   A 2-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 1 ray, 1 line,  (-1, 0, -1)),
 ((1, -1, 1),   A 3-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 2 rays, 1 line, (1, 2, 0)),
 ((1, -1, -1),  A 3-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 2 rays, 1 line, (-2, 0, -1)),
 ((-1, 0, -1),  A 2-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 1 ray, 1 line,  (0, 0, 1)),
 ((-1, 1, 0),   A 2-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 1 ray, 1 line,  (1, 0, 1)),
 ((-1, 1, 1),   A 3-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 2 rays, 1 line, (0, -2, -1)),
 ((-1, 1, -1),  A 3-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 2 rays, 1 line, (1, 0, 2)),
 ((-1, -1, -1), A 3-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 2 rays, 1 line, (-1, 0, 1))]

Python

>>> from sage.all import *
>>> # needs sage.graphs
>>> a = hyperplane_arrangements.braid(Integer(2))
>>> a.hyperplanes()
(Hyperplane t0 - t1 + 0,)
>>> a.closed_faces()
(((0,),  A 1-dimensional polyhedron in QQ^2 defined
         as the convex hull of 1 vertex and 1 line),
 ((1,),  A 2-dimensional polyhedron in QQ^2 defined
         as the convex hull of 1 vertex, 1 ray, 1 line),
 ((-1,), A 2-dimensional polyhedron in QQ^2 defined
         as the convex hull of 1 vertex, 1 ray, 1 line))
>>> a.closed_faces(labelled=False)
(A 1-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex and 1 line,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex, 1 ray, 1 line,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex, 1 ray, 1 line)
>>> [(v, F, F.representative_point()) for v, F in a.closed_faces()]
[((0,),  A 1-dimensional polyhedron in QQ^2 defined
         as the convex hull of 1 vertex and 1 line,      (0, 0)),
 ((1,),  A 2-dimensional polyhedron in QQ^2 defined
         as the convex hull of 1 vertex, 1 ray, 1 line,  (0, -1)),
 ((-1,), A 2-dimensional polyhedron in QQ^2 defined
         as the convex hull of 1 vertex, 1 ray, 1 line,  (-1, 0))]

>>> H = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> a = H(x, y+Integer(1))
>>> a.hyperplanes()
(Hyperplane 0*x + y + 1, Hyperplane x + 0*y + 0)
>>> [(v, F, F.representative_point()) for v, F in a.closed_faces()]
[((0, 0),   A 0-dimensional polyhedron in QQ^2 defined
            as the convex hull of 1 vertex,             (0, -1)),
 ((0, 1),   A 1-dimensional polyhedron in QQ^2 defined
            as the convex hull of 1 vertex and 1 ray,   (1, -1)),
 ((0, -1),  A 1-dimensional polyhedron in QQ^2 defined
            as the convex hull of 1 vertex and 1 ray,   (-1, -1)),
 ((1, 0),   A 1-dimensional polyhedron in QQ^2 defined
            as the convex hull of 1 vertex and 1 ray,   (0, 0)),
 ((1, 1),   A 2-dimensional polyhedron in QQ^2 defined
            as the convex hull of 1 vertex and 2 rays,  (1, 0)),
 ((1, -1),  A 2-dimensional polyhedron in QQ^2 defined
            as the convex hull of 1 vertex and 2 rays,  (-1, 0)),
 ((-1, 0),  A 1-dimensional polyhedron in QQ^2 defined
            as the convex hull of 1 vertex and 1 ray,   (0, -2)),
 ((-1, 1),  A 2-dimensional polyhedron in QQ^2 defined
            as the convex hull of 1 vertex and 2 rays,  (1, -2)),
 ((-1, -1), A 2-dimensional polyhedron in QQ^2 defined
            as the convex hull of 1 vertex and 2 rays,  (-1, -2))]

>>> # needs sage.graphs
>>> a = hyperplane_arrangements.braid(Integer(3))
>>> a.hyperplanes()
(Hyperplane 0*t0 + t1 - t2 + 0,
 Hyperplane t0 - t1 + 0*t2 + 0,
 Hyperplane t0 + 0*t1 - t2 + 0)
>>> [(v, F, F.representative_point()) for v, F in a.closed_faces()]
[((0, 0, 0),    A 1-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex and 1 line,      (0, 0, 0)),
 ((0, 1, 1),    A 2-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 1 ray, 1 line,  (0, -1, -1)),
 ((0, -1, -1),  A 2-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 1 ray, 1 line,  (-1, 0, 0)),
 ((1, 0, 1),    A 2-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 1 ray, 1 line,  (1, 1, 0)),
 ((1, 1, 1),    A 3-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 2 rays, 1 line, (0, -1, -2)),
 ((1, -1, 0),   A 2-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 1 ray, 1 line,  (-1, 0, -1)),
 ((1, -1, 1),   A 3-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 2 rays, 1 line, (1, 2, 0)),
 ((1, -1, -1),  A 3-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 2 rays, 1 line, (-2, 0, -1)),
 ((-1, 0, -1),  A 2-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 1 ray, 1 line,  (0, 0, 1)),
 ((-1, 1, 0),   A 2-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 1 ray, 1 line,  (1, 0, 1)),
 ((-1, 1, 1),   A 3-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 2 rays, 1 line, (0, -2, -1)),
 ((-1, 1, -1),  A 3-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 2 rays, 1 line, (1, 0, 2)),
 ((-1, -1, -1), A 3-dimensional polyhedron in QQ^3 defined
                as the convex hull of 1 vertex, 2 rays, 1 line, (-1, 0, 1))]

Sage Live

# needs sage.graphs
a = hyperplane_arrangements.braid(2)
a.hyperplanes()
a.closed_faces()
a.closed_faces(labelled=False)
[(v, F, F.representative_point()) for v, F in a.closed_faces()]
H.<x,y> = HyperplaneArrangements(QQ)
a = H(x, y+1)
a.hyperplanes()
[(v, F, F.representative_point()) for v, F in a.closed_faces()]
# needs sage.graphs
a = hyperplane_arrangements.braid(3)
a.hyperplanes()
[(v, F, F.representative_point()) for v, F in a.closed_faces()]

Let us check that the number of closed faces with a given dimension computed using self.closed_faces() equals the one computed using face_vector():

Sage

sage: def test_number(a):
....:     Qx = PolynomialRing(QQ, 'x'); x = Qx.gen()
....:     RHS = Qx.sum(vi * x ** i for i, vi in enumerate(a.face_vector()))
....:     LHS = Qx.sum(x ** F[1].dim() for F in a.closed_faces())
....:     return LHS == RHS
sage: a = hyperplane_arrangements.Catalan(2)
sage: test_number(a)                                                        # needs sage.combinat
True
sage: a = hyperplane_arrangements.Shi(3)
sage: test_number(a)                # long time                             # needs sage.combinat
True

Python

>>> from sage.all import *
>>> def test_number(a):
...     Qx = PolynomialRing(QQ, 'x'); x = Qx.gen()
...     RHS = Qx.sum(vi * x ** i for i, vi in enumerate(a.face_vector()))
...     LHS = Qx.sum(x ** F[Integer(1)].dim() for F in a.closed_faces())
...     return LHS == RHS
>>> a = hyperplane_arrangements.Catalan(Integer(2))
>>> test_number(a)                                                        # needs sage.combinat
True
>>> a = hyperplane_arrangements.Shi(Integer(3))
>>> test_number(a)                # long time                             # needs sage.combinat
True

Sage Live

def test_number(a):
    Qx = PolynomialRing(QQ, 'x'); x = Qx.gen()
    RHS = Qx.sum(vi * x ** i for i, vi in enumerate(a.face_vector()))
    LHS = Qx.sum(x ** F[1].dim() for F in a.closed_faces())
    return LHS == RHS
a = hyperplane_arrangements.Catalan(2)
test_number(a)                                                        # needs sage.combinat
a = hyperplane_arrangements.Shi(3)
test_number(a)                # long time                             # needs sage.combinat

cocharacteristic_polynomial()[source]¶

Return the cocharacteristic polynomial of self.

The cocharacteristic polynomial of a hyperplane arrangement \(A\) is defined by

\[\Psi_A(z) := \sum_{X \in L} |\mu(B,X)| z^{\dim X},\]

where \(L\) is the intersection poset of \(A\), \(B\) is the minimal element of \(L\) (here, the \(0\) dimensional subspace), and \(\mu\) is the Möbius function of \(L\).

OUTPUT: the cocharacteristic polynomial in \(\ZZ[z]\)

EXAMPLES:

Sage

sage: A = hyperplane_arrangements.coordinate(2)
sage: A.cocharacteristic_polynomial()                                       # needs sage.graphs
z^2 + 2*z + 1
sage: B = hyperplane_arrangements.braid(3)
sage: B.cocharacteristic_polynomial()                                       # needs sage.graphs
2*z^3 + 3*z^2 + z

Python

>>> from sage.all import *
>>> A = hyperplane_arrangements.coordinate(Integer(2))
>>> A.cocharacteristic_polynomial()                                       # needs sage.graphs
z^2 + 2*z + 1
>>> B = hyperplane_arrangements.braid(Integer(3))
>>> B.cocharacteristic_polynomial()                                       # needs sage.graphs
2*z^3 + 3*z^2 + z

Sage Live

A = hyperplane_arrangements.coordinate(2)
A.cocharacteristic_polynomial()                                       # needs sage.graphs
B = hyperplane_arrangements.braid(3)
B.cocharacteristic_polynomial()                                       # needs sage.graphs

cone(variable='t')[source]¶

Return the cone over the hyperplane arrangement.

INPUT:

variable – string; the name of the additional variable

OUTPUT:

A new hyperplane arrangement \(L\). Its equations consist of \([0, -d, a_1, \ldots, a_n]\) for each \([d, a_1, \ldots, a_n]\) in the original arrangement and the equation \([0, 1, 0, \ldots, 0]\) (maybe not in this order).

Warning

While there is an almost-one-to-one correspondence between the hyperplanes of self and those of self.cone(), there is no guarantee that the order in which they appear in self.hyperplanes() will match the order in which their counterparts in self.cone() will appear in self.cone().hyperplanes()! This warning does not apply to ordered hyperplane arrangements.

EXAMPLES:

Sage

sage: # needs sage.combinat
sage: a.<x,y,z> = hyperplane_arrangements.semiorder(3)
sage: b = a.cone()
sage: a.characteristic_polynomial().factor()
x * (x^2 - 6*x + 12)
sage: b.characteristic_polynomial().factor()
(x - 1) * x * (x^2 - 6*x + 12)
sage: a.hyperplanes()
(Hyperplane 0*x + y - z - 1,
 Hyperplane 0*x + y - z + 1,
 Hyperplane x - y + 0*z - 1,
 Hyperplane x - y + 0*z + 1,
 Hyperplane x + 0*y - z - 1,
 Hyperplane x + 0*y - z + 1)
sage: b.hyperplanes()
(Hyperplane -t + 0*x + y - z + 0,
 Hyperplane -t + x - y + 0*z + 0,
 Hyperplane -t + x + 0*y - z + 0,
 Hyperplane t + 0*x + 0*y + 0*z + 0,
 Hyperplane t + 0*x + y - z + 0,
 Hyperplane t + x - y + 0*z + 0,
 Hyperplane t + x + 0*y - z + 0)

Python

>>> from sage.all import *
>>> # needs sage.combinat
>>> a = hyperplane_arrangements.semiorder(Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = a._first_ngens(3)
>>> b = a.cone()
>>> a.characteristic_polynomial().factor()
x * (x^2 - 6*x + 12)
>>> b.characteristic_polynomial().factor()
(x - 1) * x * (x^2 - 6*x + 12)
>>> a.hyperplanes()
(Hyperplane 0*x + y - z - 1,
 Hyperplane 0*x + y - z + 1,
 Hyperplane x - y + 0*z - 1,
 Hyperplane x - y + 0*z + 1,
 Hyperplane x + 0*y - z - 1,
 Hyperplane x + 0*y - z + 1)
>>> b.hyperplanes()
(Hyperplane -t + 0*x + y - z + 0,
 Hyperplane -t + x - y + 0*z + 0,
 Hyperplane -t + x + 0*y - z + 0,
 Hyperplane t + 0*x + 0*y + 0*z + 0,
 Hyperplane t + 0*x + y - z + 0,
 Hyperplane t + x - y + 0*z + 0,
 Hyperplane t + x + 0*y - z + 0)

Sage Live

# needs sage.combinat
a.<x,y,z> = hyperplane_arrangements.semiorder(3)
b = a.cone()
a.characteristic_polynomial().factor()
b.characteristic_polynomial().factor()
a.hyperplanes()
b.hyperplanes()

defining_polynomial()[source]¶

Return the defining polynomial of A.

Let \(A = (H_i)_i\) be a hyperplane arrangement in a vector space \(V\) corresponding to the null spaces of \(\alpha_{H_i} \in V^*\). Then the defining polynomial of \(A\) is given by

\[Q(A) = \prod_i \alpha_{H_i} \in S(V^*).\]

EXAMPLES:

Sage

sage: H.<x,y,z> = HyperplaneArrangements(QQ)
sage: A = H([2*x + y - z, -x - 2*y + z])
sage: p = A.defining_polynomial(); p
-2*x^2 - 5*x*y - 2*y^2 + 3*x*z + 3*y*z - z^2
sage: p.factor()
(-1) * (x + 2*y - z) * (2*x + y - z)

Python

>>> from sage.all import *
>>> H = HyperplaneArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = H._first_ngens(3)
>>> A = H([Integer(2)*x + y - z, -x - Integer(2)*y + z])
>>> p = A.defining_polynomial(); p
-2*x^2 - 5*x*y - 2*y^2 + 3*x*z + 3*y*z - z^2
>>> p.factor()
(-1) * (x + 2*y - z) * (2*x + y - z)

Sage Live

H.<x,y,z> = HyperplaneArrangements(QQ)
A = H([2*x + y - z, -x - 2*y + z])
p = A.defining_polynomial(); p
p.factor()

deletion(hyperplanes)[source]¶

Return the hyperplane arrangement obtained by removing h.

INPUT:

h – a hyperplane or hyperplane arrangement

OUTPUT:

A new hyperplane arrangement with the given hyperplane(s) h removed.

See also

restriction()

EXAMPLES:

Sage

sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: A = H([0,1,0], [1,0,1], [-1,0,1], [0,1,-1], [0,1,1]);  A
Arrangement of 5 hyperplanes of dimension 2 and rank 2
sage: A.deletion(x)
Arrangement <y - 1 | y + 1 | x - y | x + y>
sage: h = H([0,1,0], [0,1,1])
sage: A.deletion(h)
Arrangement <y - 1 | y + 1 | x - y>

Python

>>> from sage.all import *
>>> H = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H([Integer(0),Integer(1),Integer(0)], [Integer(1),Integer(0),Integer(1)], [-Integer(1),Integer(0),Integer(1)], [Integer(0),Integer(1),-Integer(1)], [Integer(0),Integer(1),Integer(1)]);  A
Arrangement of 5 hyperplanes of dimension 2 and rank 2
>>> A.deletion(x)
Arrangement <y - 1 | y + 1 | x - y | x + y>
>>> h = H([Integer(0),Integer(1),Integer(0)], [Integer(0),Integer(1),Integer(1)])
>>> A.deletion(h)
Arrangement <y - 1 | y + 1 | x - y>

Sage Live

H.<x,y> = HyperplaneArrangements(QQ)
A = H([0,1,0], [1,0,1], [-1,0,1], [0,1,-1], [0,1,1]);  A
A.deletion(x)
h = H([0,1,0], [0,1,1])
A.deletion(h)

derivation_module_basis(algorithm='singular')[source]¶

Return a basis for the derivation module of self if one exists, otherwise return None.

INPUT:

algorithm – (default: 'singular') can be one of the following:
- 'singular' – use Singular’s minimal free resolution
- 'BC' – use the algorithm given by Barakat and Cuntz in [BC2012] (much slower than using Singular)

OUTPUT:

A basis for the derivation module (over \(S\), the symmetric space) as vectors of a free module over \(S\).

ALGORITHM:

Singular

This gets the reduced syzygy module of the Jacobian ideal of the defining polynomial \(f\) of self. It then checks Saito’s criterion that the determinant of the basis matrix is a scalar multiple of \(f\). If the basis matrix is not square or it fails Saito’s criterion, then we check if the arrangement is free. If it is free, then we fall back to the Barakat-Cuntz algorithm.

Return the product of the derivation module free chain matrices. See Section 6 of [BC2012].

EXAMPLES:

Sage

sage: # needs sage.combinat sage.groups
sage: W = WeylGroup(['A', 2], prefix='s')
sage: A = W.long_element().inversion_arrangement()
sage: A.derivation_module_basis()
[(a1, a2), (0, a1*a2 + a2^2)]

Python

>>> from sage.all import *
>>> # needs sage.combinat sage.groups
>>> W = WeylGroup(['A', Integer(2)], prefix='s')
>>> A = W.long_element().inversion_arrangement()
>>> A.derivation_module_basis()
[(a1, a2), (0, a1*a2 + a2^2)]

Sage Live

# needs sage.combinat sage.groups
W = WeylGroup(['A', 2], prefix='s')
A = W.long_element().inversion_arrangement()
A.derivation_module_basis()

derivation_module_free_chain()[source]¶

Return a free chain for the derivation module if one exists, otherwise return None.

See also

is_free()

EXAMPLES:

Sage

sage: # needs sage.combinat sage.groups
sage: W = WeylGroup(['A',3], prefix='s')
sage: A = W.long_element().inversion_arrangement()
sage: for M in A.derivation_module_free_chain(): print("%s\n"%M)
[ 1  0  0]
[ 0  1  0]
[ 0  0 a3]

[ 1  0  0]
[ 0  0  1]
[ 0 a2  0]

[  1   0   0]
[  0  -1  -1]
[  0  a2 -a3]

[ 0  1  0]
[ 0  0  1]
[a1  0  0]

[ 1  0 -1]
[a3 -1  0]
[a1  0 a2]

[       1        0        0]
[      a3       -1       -1]
[       0       a1 -a2 - a3]

Python

>>> from sage.all import *
>>> # needs sage.combinat sage.groups
>>> W = WeylGroup(['A',Integer(3)], prefix='s')
>>> A = W.long_element().inversion_arrangement()
>>> for M in A.derivation_module_free_chain(): print("%s\n"%M)
[ 1  0  0]
[ 0  1  0]
[ 0  0 a3]
<BLANKLINE>
[ 1  0  0]
[ 0  0  1]
[ 0 a2  0]
<BLANKLINE>
[  1   0   0]
[  0  -1  -1]
[  0  a2 -a3]
<BLANKLINE>
[ 0  1  0]
[ 0  0  1]
[a1  0  0]
<BLANKLINE>
[ 1  0 -1]
[a3 -1  0]
[a1  0 a2]
<BLANKLINE>
[       1        0        0]
[      a3       -1       -1]
[       0       a1 -a2 - a3]
<BLANKLINE>

Sage Live

# needs sage.combinat sage.groups
W = WeylGroup(['A',3], prefix='s')
A = W.long_element().inversion_arrangement()
for M in A.derivation_module_free_chain(): print("%s\n"%M)

dimension()[source]¶

Return the ambient space dimension of the arrangement.

OUTPUT: integer

EXAMPLES:

Sage

sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: (x | x-1 | x+1).dimension()
2
sage: H(x).dimension()
2

Python

>>> from sage.all import *
>>> H = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> (x | x-Integer(1) | x+Integer(1)).dimension()
2
>>> H(x).dimension()
2

Sage Live

H.<x,y> = HyperplaneArrangements(QQ)
(x | x-1 | x+1).dimension()
H(x).dimension()

distance_between_regions(region1, region2)[source]¶

Return the number of hyperplanes separating the two regions.

INPUT:

region1, region2 – regions of the arrangement or representative points of regions

OUTPUT: integer; the number of hyperplanes separating the two regions

EXAMPLES:

Sage

sage: c = hyperplane_arrangements.coordinate(2)
sage: r = c.region_containing_point([-1, -1])
sage: s = c.region_containing_point([1, 1])
sage: c.distance_between_regions(r, s)
2
sage: c.distance_between_regions(s, s)
0

Python

>>> from sage.all import *
>>> c = hyperplane_arrangements.coordinate(Integer(2))
>>> r = c.region_containing_point([-Integer(1), -Integer(1)])
>>> s = c.region_containing_point([Integer(1), Integer(1)])
>>> c.distance_between_regions(r, s)
2
>>> c.distance_between_regions(s, s)
0

Sage Live

c = hyperplane_arrangements.coordinate(2)
r = c.region_containing_point([-1, -1])
s = c.region_containing_point([1, 1])
c.distance_between_regions(r, s)
c.distance_between_regions(s, s)

distance_enumerator(base_region)[source]¶

Return the generating function for the number of hyperplanes at given distance.

INPUT:

base_region – region of arrangement or point in region

OUTPUT:

A polynomial \(f(x)\) for which the coefficient of \(x^i\) is the number of hyperplanes of distance \(i\) from base_region, i.e., the number of hyperplanes separated by \(i\) hyperplanes from base_region.

EXAMPLES:

Sage

sage: c = hyperplane_arrangements.coordinate(3)
sage: c.distance_enumerator(c.region_containing_point([1,1,1]))
x^3 + 3*x^2 + 3*x + 1

Python

>>> from sage.all import *
>>> c = hyperplane_arrangements.coordinate(Integer(3))
>>> c.distance_enumerator(c.region_containing_point([Integer(1),Integer(1),Integer(1)]))
x^3 + 3*x^2 + 3*x + 1

Sage Live

c = hyperplane_arrangements.coordinate(3)
c.distance_enumerator(c.region_containing_point([1,1,1]))

doubly_indexed_whitney_number(i, j, kind=1)[source]¶

Return the \(i,j\)-th doubly-indexed Whitney number.

If kind=1, this number is obtained by adding the Möbius function values \(mu(x,y)\) over all \(x, y\) in the intersection poset with \(\mathrm{rank}(x) = i\) and \(\mathrm{rank}(y) = j\).

If \(kind=2\), this number is the number of elements \(x,y\) in the intersection poset such that \(x \leq y\) with ranks \(i\) and \(j\), respectively.

INPUT:

i, j – integers
kind – (default: 1) 1 or 2

OUTPUT:

Integer. The \((i,j)\)-th entry of the kind Whitney number.

See also

whitney_number(), whitney_data()

EXAMPLES:

Sage

sage: # needs sage.combinat
sage: A = hyperplane_arrangements.Shi(3)
sage: A.doubly_indexed_whitney_number(0, 2)
9
sage: A.whitney_number(2)
9
sage: A.doubly_indexed_whitney_number(1, 2)
-15

Python

>>> from sage.all import *
>>> # needs sage.combinat
>>> A = hyperplane_arrangements.Shi(Integer(3))
>>> A.doubly_indexed_whitney_number(Integer(0), Integer(2))
9
>>> A.whitney_number(Integer(2))
9
>>> A.doubly_indexed_whitney_number(Integer(1), Integer(2))
-15

Sage Live

# needs sage.combinat
A = hyperplane_arrangements.Shi(3)
A.doubly_indexed_whitney_number(0, 2)
A.whitney_number(2)
A.doubly_indexed_whitney_number(1, 2)

REFERENCES:

[GZ1983]

essentialization()[source]¶

Return the essentialization of the hyperplane arrangement.

The essentialization of a hyperplane arrangement whose base field has characteristic 0 is obtained by intersecting the hyperplanes by the space spanned by their normal vectors.

OUTPUT:

The essentialization \(\mathcal{A}'\) of \(\mathcal{A}\) as a new hyperplane arrangement.

EXAMPLES:

Sage

sage: a = hyperplane_arrangements.braid(3)                                  # needs sage.graphs
sage: a.is_essential()                                                      # needs sage.graphs
False
sage: a.essentialization()                                                  # needs sage.graphs
Arrangement <t1 - t2 | t1 + 2*t2 | 2*t1 + t2>

sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: B = H([(1,0),1], [(1,0),-1])
sage: B.is_essential()
False
sage: B.essentialization()
Arrangement <-x + 1 | x + 1>
sage: B.essentialization().parent()
Hyperplane arrangements in 1-dimensional linear space over
Rational Field with coordinate x

sage: H.<x,y> = HyperplaneArrangements(GF(2))
sage: C = H([(1,1),1], [(1,1),0])
sage: C.essentialization()
Arrangement <y | y + 1>

sage: h = hyperplane_arrangements.semiorder(4)
sage: h.essentialization()
Arrangement of 12 hyperplanes of dimension 3 and rank 3

Python

>>> from sage.all import *
>>> a = hyperplane_arrangements.braid(Integer(3))                                  # needs sage.graphs
>>> a.is_essential()                                                      # needs sage.graphs
False
>>> a.essentialization()                                                  # needs sage.graphs
Arrangement <t1 - t2 | t1 + 2*t2 | 2*t1 + t2>

>>> H = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> B = H([(Integer(1),Integer(0)),Integer(1)], [(Integer(1),Integer(0)),-Integer(1)])
>>> B.is_essential()
False
>>> B.essentialization()
Arrangement <-x + 1 | x + 1>
>>> B.essentialization().parent()
Hyperplane arrangements in 1-dimensional linear space over
Rational Field with coordinate x

>>> H = HyperplaneArrangements(GF(Integer(2)), names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> C = H([(Integer(1),Integer(1)),Integer(1)], [(Integer(1),Integer(1)),Integer(0)])
>>> C.essentialization()
Arrangement <y | y + 1>

>>> h = hyperplane_arrangements.semiorder(Integer(4))
>>> h.essentialization()
Arrangement of 12 hyperplanes of dimension 3 and rank 3

Sage Live

a = hyperplane_arrangements.braid(3)                                  # needs sage.graphs
a.is_essential()                                                      # needs sage.graphs
a.essentialization()                                                  # needs sage.graphs
H.<x,y> = HyperplaneArrangements(QQ)
B = H([(1,0),1], [(1,0),-1])
B.is_essential()
B.essentialization()
B.essentialization().parent()
H.<x,y> = HyperplaneArrangements(GF(2))
C = H([(1,1),1], [(1,1),0])
C.essentialization()
h = hyperplane_arrangements.semiorder(4)
h.essentialization()

face_product(F, G, normalize=True)[source]¶

Return the product \(FG\) in the face semigroup of self, where \(F\) and \(G\) are two closed faces of self.

The face semigroup of a hyperplane arrangement \(\mathcal{A}\) is defined as follows: As a set, it is the set of all open faces of self (see closed_faces()). Its product is defined by the following rule: If \(F\) and \(G\) are two open faces of \(\mathcal{A}\), then \(FG\) is an open face of \(\mathcal{A}\), and for every hyperplane \(H \in \mathcal{A}\), the open face \(FG\) lies on the same side of \(H\) as \(F\) unless \(F \subseteq H\), in which case \(FG\) lies on the same side of \(H\) as \(G\). Alternatively, \(FG\) can be defined as follows: If \(f\) and \(g\) are two points in \(F\) and \(G\), respectively, then \(FG\) is the face that contains the point \((f + \varepsilon g) / (1 + \varepsilon)\) for any sufficiently small positive \(\varepsilon\).

In our implementation, the face semigroup consists of closed faces rather than open faces (thanks to the 1-to-1 correspondence between open faces and closed faces, this is not really a different semigroup); these closed faces are given as polyhedra.

The face semigroup of a hyperplane arrangement is always a left-regular band (i.e., a semigroup satisfying the identities \(x^2 = x\) and \(xyx = xy\)). When the arrangement is central, then this semigroup is a monoid. See [Br2000] (Appendix A in particular) for further properties.

INPUT:

F, G – two faces of self (as polyhedra)
normalize – boolean (default: True); if True, then this method returns the precise instance of \(FG\) in the list returned by self.closed_faces(), rather than creating a new instance

EXAMPLES:

Sage

sage: # needs sage.graphs
sage: a = hyperplane_arrangements.braid(3)
sage: a.hyperplanes()
(Hyperplane 0*t0 + t1 - t2 + 0,
 Hyperplane t0 - t1 + 0*t2 + 0,
 Hyperplane t0 + 0*t1 - t2 + 0)
sage: faces = {F0: F1 for F0, F1 in a.closed_faces()}
sage: xGyEz = faces[(0, 1, 1)]   # closed face x >= y = z
sage: xGyEz.representative_point()
(0, -1, -1)
sage: xGyEz = faces[(0, 1, 1)]   # closed face x >= y = z
sage: xGyEz.representative_point()
(0, -1, -1)
sage: yGxGz = faces[(1, -1, 1)]  # closed face y >= x >= z
sage: xGyGz = faces[(1, 1, 1)]   # closed face x >= y >= z
sage: a.face_product(xGyEz, yGxGz) == xGyGz
True
sage: a.face_product(yGxGz, xGyEz) == yGxGz
True
sage: xEzGy = faces[(-1, 1, 0)]  # closed face x = z >= y
sage: xGzGy = faces[(-1, 1, 1)]  # closed face x >= z >= y
sage: a.face_product(xEzGy, yGxGz) == xGzGy
True

Python

>>> from sage.all import *
>>> # needs sage.graphs
>>> a = hyperplane_arrangements.braid(Integer(3))
>>> a.hyperplanes()
(Hyperplane 0*t0 + t1 - t2 + 0,
 Hyperplane t0 - t1 + 0*t2 + 0,
 Hyperplane t0 + 0*t1 - t2 + 0)
>>> faces = {F0: F1 for F0, F1 in a.closed_faces()}
>>> xGyEz = faces[(Integer(0), Integer(1), Integer(1))]   # closed face x >= y = z
>>> xGyEz.representative_point()
(0, -1, -1)
>>> xGyEz = faces[(Integer(0), Integer(1), Integer(1))]   # closed face x >= y = z
>>> xGyEz.representative_point()
(0, -1, -1)
>>> yGxGz = faces[(Integer(1), -Integer(1), Integer(1))]  # closed face y >= x >= z
>>> xGyGz = faces[(Integer(1), Integer(1), Integer(1))]   # closed face x >= y >= z
>>> a.face_product(xGyEz, yGxGz) == xGyGz
True
>>> a.face_product(yGxGz, xGyEz) == yGxGz
True
>>> xEzGy = faces[(-Integer(1), Integer(1), Integer(0))]  # closed face x = z >= y
>>> xGzGy = faces[(-Integer(1), Integer(1), Integer(1))]  # closed face x >= z >= y
>>> a.face_product(xEzGy, yGxGz) == xGzGy
True

Sage Live

# needs sage.graphs
a = hyperplane_arrangements.braid(3)
a.hyperplanes()
faces = {F0: F1 for F0, F1 in a.closed_faces()}
xGyEz = faces[(0, 1, 1)]   # closed face x >= y = z
xGyEz.representative_point()
xGyEz = faces[(0, 1, 1)]   # closed face x >= y = z
xGyEz.representative_point()
yGxGz = faces[(1, -1, 1)]  # closed face y >= x >= z
xGyGz = faces[(1, 1, 1)]   # closed face x >= y >= z
a.face_product(xGyEz, yGxGz) == xGyGz
a.face_product(yGxGz, xGyEz) == yGxGz
xEzGy = faces[(-1, 1, 0)]  # closed face x = z >= y
xGzGy = faces[(-1, 1, 1)]  # closed face x >= z >= y
a.face_product(xEzGy, yGxGz) == xGzGy

face_semigroup_algebra(field=None, names='e')[source]¶

Return the face semigroup algebra of self.

This is the semigroup algebra of the face semigroup of self (see face_product() for the definition of the semigroup).

Due to limitations of the current Sage codebase (e.g., semigroup algebras do not profit from the functionality of the FiniteDimensionalAlgebra class), this is implemented not as a semigroup algebra, but as a FiniteDimensionalAlgebra. The closed faces of self (in the order in which the closed_faces() method outputs them) are identified with the vectors \((0, 0, \ldots, 0, 1, 0, 0, \ldots, 0)\) (with the \(1\) moving from left to right).

INPUT:

field – a field (default: \(\QQ\)), to be used as the base ring for the algebra (can also be a commutative ring, but then certain representation-theoretical methods might misbehave)
names – (default: 'e') string; names for the basis elements of the algebra

Todo

Also implement it as an actual semigroup algebra?

EXAMPLES:

Sage

sage: # needs sage.graphs
sage: a = hyperplane_arrangements.braid(3)
sage: [(i, F[0]) for i, F in enumerate(a.closed_faces())]
[(0, (0, 0, 0)),
 (1, (0, 1, 1)),
 (2, (0, -1, -1)),
 (3, (1, 0, 1)),
 (4, (1, 1, 1)),
 (5, (1, -1, 0)),
 (6, (1, -1, 1)),
 (7, (1, -1, -1)),
 (8, (-1, 0, -1)),
 (9, (-1, 1, 0)),
 (10, (-1, 1, 1)),
 (11, (-1, 1, -1)),
 (12, (-1, -1, -1))]
sage: U = a.face_semigroup_algebra(); U
Finite-dimensional algebra of degree 13 over Rational Field
sage: e0, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10, e11, e12 = U.basis()
sage: e0 * e1
e1
sage: e0 * e5
e5
sage: e5 * e0
e5
sage: e3 * e2
e6
sage: e7 * e12
e7
sage: e3 * e12
e6
sage: e4 * e8
e4
sage: e8 * e4
e11
sage: e8 * e1
e11
sage: e5 * e12
e7
sage: (e3 + 2*e4) * (e1 - e7)
e4 - e6

sage: U3 = a.face_semigroup_algebra(field=GF(3)); U3                        # needs sage.graphs sage.rings.finite_rings
Finite-dimensional algebra of degree 13 over Finite Field of size 3

Python

>>> from sage.all import *
>>> # needs sage.graphs
>>> a = hyperplane_arrangements.braid(Integer(3))
>>> [(i, F[Integer(0)]) for i, F in enumerate(a.closed_faces())]
[(0, (0, 0, 0)),
 (1, (0, 1, 1)),
 (2, (0, -1, -1)),
 (3, (1, 0, 1)),
 (4, (1, 1, 1)),
 (5, (1, -1, 0)),
 (6, (1, -1, 1)),
 (7, (1, -1, -1)),
 (8, (-1, 0, -1)),
 (9, (-1, 1, 0)),
 (10, (-1, 1, 1)),
 (11, (-1, 1, -1)),
 (12, (-1, -1, -1))]
>>> U = a.face_semigroup_algebra(); U
Finite-dimensional algebra of degree 13 over Rational Field
>>> e0, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10, e11, e12 = U.basis()
>>> e0 * e1
e1
>>> e0 * e5
e5
>>> e5 * e0
e5
>>> e3 * e2
e6
>>> e7 * e12
e7
>>> e3 * e12
e6
>>> e4 * e8
e4
>>> e8 * e4
e11
>>> e8 * e1
e11
>>> e5 * e12
e7
>>> (e3 + Integer(2)*e4) * (e1 - e7)
e4 - e6

>>> U3 = a.face_semigroup_algebra(field=GF(Integer(3))); U3                        # needs sage.graphs sage.rings.finite_rings
Finite-dimensional algebra of degree 13 over Finite Field of size 3

Sage Live

# needs sage.graphs
a = hyperplane_arrangements.braid(3)
[(i, F[0]) for i, F in enumerate(a.closed_faces())]
U = a.face_semigroup_algebra(); U
e0, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10, e11, e12 = U.basis()
e0 * e1
e0 * e5
e5 * e0
e3 * e2
e7 * e12
e3 * e12
e4 * e8
e8 * e4
e8 * e1
e5 * e12
(e3 + 2*e4) * (e1 - e7)
U3 = a.face_semigroup_algebra(field=GF(3)); U3                        # needs sage.graphs sage.rings.finite_rings

face_vector()[source]¶

Return the face vector.

OUTPUT: a vector of integers

The \(d\)-th entry is the number of faces of dimension \(d\). A face is the intersection of a region with a hyperplane of the arrangement.

EXAMPLES:

Sage

sage: A = hyperplane_arrangements.Shi(3)
sage: A.face_vector()                                                       # needs sage.combinat
(0, 6, 21, 16)

Python

>>> from sage.all import *
>>> A = hyperplane_arrangements.Shi(Integer(3))
>>> A.face_vector()                                                       # needs sage.combinat
(0, 6, 21, 16)

Sage Live

A = hyperplane_arrangements.Shi(3)
A.face_vector()                                                       # needs sage.combinat

has_good_reduction(p)[source]¶

Return whether the hyperplane arrangement has good reduction mod \(p\).

Let \(A\) be a hyperplane arrangement with equations defined over the integers, and let \(B\) be the hyperplane arrangement defined by reducing these equations modulo a prime \(p\). Then \(A\) has good reduction modulo \(p\) if the intersection posets of \(A\) and \(B\) are isomorphic.

INPUT:

p – prime number

OUTPUT: boolean

EXAMPLES:

Sage

sage: # needs sage.combinat
sage: a = hyperplane_arrangements.semiorder(3)
sage: a.has_good_reduction(5)
True
sage: a.has_good_reduction(3)
False
sage: b = a.change_ring(GF(3))
sage: a.characteristic_polynomial()
x^3 - 6*x^2 + 12*x
sage: b.characteristic_polynomial()  # not equal to that for a
x^3 - 6*x^2 + 10*x

Python

>>> from sage.all import *
>>> # needs sage.combinat
>>> a = hyperplane_arrangements.semiorder(Integer(3))
>>> a.has_good_reduction(Integer(5))
True
>>> a.has_good_reduction(Integer(3))
False
>>> b = a.change_ring(GF(Integer(3)))
>>> a.characteristic_polynomial()
x^3 - 6*x^2 + 12*x
>>> b.characteristic_polynomial()  # not equal to that for a
x^3 - 6*x^2 + 10*x

Sage Live

# needs sage.combinat
a = hyperplane_arrangements.semiorder(3)
a.has_good_reduction(5)
a.has_good_reduction(3)
b = a.change_ring(GF(3))
a.characteristic_polynomial()
b.characteristic_polynomial()  # not equal to that for a

hyperplanes()[source]¶

Return the hyperplanes in the arrangement as a tuple.

OUTPUT: a tuple

EXAMPLES:

Sage

sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: A = H([1,1,0], [2,3,-1], [4,5,3])
sage: A.hyperplanes()
(Hyperplane x + 0*y + 1, Hyperplane 3*x - y + 2, Hyperplane 5*x + 3*y + 4)

Python

>>> from sage.all import *
>>> H = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H([Integer(1),Integer(1),Integer(0)], [Integer(2),Integer(3),-Integer(1)], [Integer(4),Integer(5),Integer(3)])
>>> A.hyperplanes()
(Hyperplane x + 0*y + 1, Hyperplane 3*x - y + 2, Hyperplane 5*x + 3*y + 4)

Sage Live

H.<x,y> = HyperplaneArrangements(QQ)
A = H([1,1,0], [2,3,-1], [4,5,3])
A.hyperplanes()

Note that the hyperplanes can be indexed as if they were a list:

Sage

sage: A[0]
Hyperplane x + 0*y + 1

Python

>>> from sage.all import *
>>> A[Integer(0)]
Hyperplane x + 0*y + 1

Sage Live

A[0]

intersection_poset(element_label='int')[source]¶

Return the intersection poset of the hyperplane arrangement.

INPUT:

element_label – (default: 'int') specify how an intersection should be represented; must be one of the following:
- 'subspace' – as a subspace
- 'subset' – as a subset of the defining hyperplanes
- 'int' – as an integer

OUTPUT:

The poset of non-empty intersections of hyperplanes, with intersections represented by integers, subsets of integers or subspaces (see the examples for more details).

EXAMPLES:

By default, the elements of the poset are the integers from \(0\) through the cardinality of the poset minus one. The element labelled \(0\) always corresponds to the ambient vector space, and the hyperplanes themselves are labelled \(1, 2, \ldots, n\), where \(n\) is the number of hyperplanes of the arrangement.

Sage

sage: A = hyperplane_arrangements.coordinate(2)
sage: L = A.intersection_poset(); L                                         # needs sage.combinat
Finite poset containing 4 elements
sage: sorted(L)                                                             # needs sage.combinat
[0, 1, 2, 3]
sage: L.level_sets()                                                        # needs sage.combinat
[[0], [1, 2], [3]]

Python

>>> from sage.all import *
>>> A = hyperplane_arrangements.coordinate(Integer(2))
>>> L = A.intersection_poset(); L                                         # needs sage.combinat
Finite poset containing 4 elements
>>> sorted(L)                                                             # needs sage.combinat
[0, 1, 2, 3]
>>> L.level_sets()                                                        # needs sage.combinat
[[0], [1, 2], [3]]

Sage Live

A = hyperplane_arrangements.coordinate(2)
L = A.intersection_poset(); L                                         # needs sage.combinat
sorted(L)                                                             # needs sage.combinat
L.level_sets()                                                        # needs sage.combinat

Sage

sage: # needs sage.combinat
sage: A = hyperplane_arrangements.semiorder(3)
sage: L = A.intersection_poset(); L
Finite poset containing 19 elements
sage: sorted(L)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]
sage: [sorted(level_set) for level_set in L.level_sets()]
[[0], [1, 2, 3, 4, 5, 6], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]]

Python

>>> from sage.all import *
>>> # needs sage.combinat
>>> A = hyperplane_arrangements.semiorder(Integer(3))
>>> L = A.intersection_poset(); L
Finite poset containing 19 elements
>>> sorted(L)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]
>>> [sorted(level_set) for level_set in L.level_sets()]
[[0], [1, 2, 3, 4, 5, 6], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]]

Sage Live

# needs sage.combinat
A = hyperplane_arrangements.semiorder(3)
L = A.intersection_poset(); L
sorted(L)
[sorted(level_set) for level_set in L.level_sets()]

Python

>>> from sage.all import *
>>> # needs sage.combinat
>>> A = hyperplane_arrangements.semiorder(Integer(3))
>>> L = A.intersection_poset(); L
Finite poset containing 19 elements
>>> sorted(L)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]
>>> [sorted(level_set) for level_set in L.level_sets()]
[[0], [1, 2, 3, 4, 5, 6], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]]

Sage Live

# needs sage.combinat
A = hyperplane_arrangements.semiorder(3)
L = A.intersection_poset(); L
sorted(L)
[sorted(level_set) for level_set in L.level_sets()]

By passing the argument element_label="subset", each element of the intersection poset is labelled by the set of indices of the hyperplanes whose intersection is said element. The index of a hyperplane is its index in self.hyperplanes().

Sage

sage: A = hyperplane_arrangements.semiorder(3)
sage: L = A.intersection_poset(element_label='subset')                      # needs sage.combinat
sage: [sorted(level, key=sorted) for level in L.level_sets()]               # needs sage.combinat
[[{}],
 [{0}, {1}, {2}, {3}, {4}, {5}],
 [{0, 2}, {0, 3}, {0, 4}, {0, 5}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 4}, {2, 5}, {3, 4}, {3, 5}]]

Python

>>> from sage.all import *
>>> A = hyperplane_arrangements.semiorder(Integer(3))
>>> L = A.intersection_poset(element_label='subset')                      # needs sage.combinat
>>> [sorted(level, key=sorted) for level in L.level_sets()]               # needs sage.combinat
[[{}],
 [{0}, {1}, {2}, {3}, {4}, {5}],
 [{0, 2}, {0, 3}, {0, 4}, {0, 5}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 4}, {2, 5}, {3, 4}, {3, 5}]]

Sage Live

A = hyperplane_arrangements.semiorder(3)
L = A.intersection_poset(element_label='subset')                      # needs sage.combinat
[sorted(level, key=sorted) for level in L.level_sets()]               # needs sage.combinat

Sage

sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: A = H((y, y-1, y+1, x-y, x+y))
sage: L = A.intersection_poset(element_label='subset')                      # needs sage.combinat
sage: sorted(L, key=sorted)                                                 # needs sage.combinat
[{}, {0}, {0, 3}, {0, 4}, {1}, {1, 3, 4}, {2}, {2, 3}, {2, 4}, {3}, {4}]

Python

>>> from sage.all import *
>>> H = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H((y, y-Integer(1), y+Integer(1), x-y, x+y))
>>> L = A.intersection_poset(element_label='subset')                      # needs sage.combinat
>>> sorted(L, key=sorted)                                                 # needs sage.combinat
[{}, {0}, {0, 3}, {0, 4}, {1}, {1, 3, 4}, {2}, {2, 3}, {2, 4}, {3}, {4}]

Sage Live

H.<x,y> = HyperplaneArrangements(QQ)
A = H((y, y-1, y+1, x-y, x+y))
L = A.intersection_poset(element_label='subset')                      # needs sage.combinat
sorted(L, key=sorted)                                                 # needs sage.combinat

Python

>>> from sage.all import *
>>> H = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H((y, y-Integer(1), y+Integer(1), x-y, x+y))
>>> L = A.intersection_poset(element_label='subset')                      # needs sage.combinat
>>> sorted(L, key=sorted)                                                 # needs sage.combinat
[{}, {0}, {0, 3}, {0, 4}, {1}, {1, 3, 4}, {2}, {2, 3}, {2, 4}, {3}, {4}]

Sage Live

H.<x,y> = HyperplaneArrangements(QQ)
A = H((y, y-1, y+1, x-y, x+y))
L = A.intersection_poset(element_label='subset')                      # needs sage.combinat
sorted(L, key=sorted)                                                 # needs sage.combinat

One can instead use affine subspaces as elements, which is what is used to compute the poset in the first place:

Sage

sage: A = hyperplane_arrangements.coordinate(2)
sage: L = A.intersection_poset(element_label='subspace'); L                 # needs sage.combinat
Finite poset containing 4 elements
sage: sorted(L, key=lambda S: (S.dimension(),                               # needs sage.combinat
....:                          S.linear_part().basis_matrix()))
[Affine space p + W where:
   p = (0, 0)
   W = Vector space of degree 2 and dimension 0 over Rational Field
       Basis matrix: [],
 Affine space p + W where:
   p = (0, 0)
   W = Vector space of degree 2 and dimension 1 over Rational Field
       Basis matrix: [0 1],
 Affine space p + W where:
   p = (0, 0)
   W = Vector space of degree 2 and dimension 1 over Rational Field
       Basis matrix: [1 0],
 Affine space p + W where:
   p = (0, 0)
   W = Vector space of dimension 2 over Rational Field]

Python

>>> from sage.all import *
>>> A = hyperplane_arrangements.coordinate(Integer(2))
>>> L = A.intersection_poset(element_label='subspace'); L                 # needs sage.combinat
Finite poset containing 4 elements
>>> sorted(L, key=lambda S: (S.dimension(),                               # needs sage.combinat
...                          S.linear_part().basis_matrix()))
[Affine space p + W where:
   p = (0, 0)
   W = Vector space of degree 2 and dimension 0 over Rational Field
       Basis matrix: [],
 Affine space p + W where:
   p = (0, 0)
   W = Vector space of degree 2 and dimension 1 over Rational Field
       Basis matrix: [0 1],
 Affine space p + W where:
   p = (0, 0)
   W = Vector space of degree 2 and dimension 1 over Rational Field
       Basis matrix: [1 0],
 Affine space p + W where:
   p = (0, 0)
   W = Vector space of dimension 2 over Rational Field]

Sage Live

A = hyperplane_arrangements.coordinate(2)
L = A.intersection_poset(element_label='subspace'); L                 # needs sage.combinat
sorted(L, key=lambda S: (S.dimension(),                               # needs sage.combinat
                         S.linear_part().basis_matrix()))

is_central(certificate=False)[source]¶

Test whether the intersection of all the hyperplanes is nonempty.

A hyperplane arrangement is central if the intersection of all the hyperplanes in the arrangement is nonempty.

INPUT:

certificate – boolean (default: False); specifies whether to return the center as a polyhedron (possibly empty) as part of the output

OUTPUT: if certificate is True, returns a tuple containing:

A boolean
The polyhedron defined to be the intersection of all the hyperplanes

If certificate is False, returns a boolean.

EXAMPLES:

Sage

sage: a = hyperplane_arrangements.braid(2)                                  # needs sage.graphs
sage: a.is_central()                                                        # needs sage.graphs
True

Python

>>> from sage.all import *
>>> a = hyperplane_arrangements.braid(Integer(2))                                  # needs sage.graphs
>>> a.is_central()                                                        # needs sage.graphs
True

Sage Live

a = hyperplane_arrangements.braid(2)                                  # needs sage.graphs
a.is_central()                                                        # needs sage.graphs

The Catalan arrangement in dimension 3 is not central:

Sage

sage: b = hyperplane_arrangements.Catalan(3)
sage: b.is_central(certificate=True)
(False, The empty polyhedron in QQ^3)

Python

>>> from sage.all import *
>>> b = hyperplane_arrangements.Catalan(Integer(3))
>>> b.is_central(certificate=True)
(False, The empty polyhedron in QQ^3)

Sage Live

b = hyperplane_arrangements.Catalan(3)
b.is_central(certificate=True)

The empty arrangement in dimension 5 is central:

Sage

sage: H = HyperplaneArrangements(QQ, names=tuple(['x'+str(i) for i in range(7)]))
sage: c = H()
sage: c.is_central(certificate=True)
(True, A 7-dimensional polyhedron in QQ^7 defined
       as the convex hull of 1 vertex and 7 lines)

Python

>>> from sage.all import *
>>> H = HyperplaneArrangements(QQ, names=tuple(['x'+str(i) for i in range(Integer(7))]))
>>> c = H()
>>> c.is_central(certificate=True)
(True, A 7-dimensional polyhedron in QQ^7 defined
       as the convex hull of 1 vertex and 7 lines)

Sage Live

H = HyperplaneArrangements(QQ, names=tuple(['x'+str(i) for i in range(7)]))
c = H()
c.is_central(certificate=True)

is_essential()[source]¶

Test whether the hyperplane arrangement is essential.

A hyperplane arrangement is essential if the span of the normals of its hyperplanes spans the ambient space.

See also

essentialization()

OUTPUT: boolean

EXAMPLES:

Sage

sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: H(x, x+1).is_essential()
False
sage: H(x, y).is_essential()
True

Python

>>> from sage.all import *
>>> H = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> H(x, x+Integer(1)).is_essential()
False
>>> H(x, y).is_essential()
True

Sage Live

H.<x,y> = HyperplaneArrangements(QQ)
H(x, x+1).is_essential()
H(x, y).is_essential()

is_formal()[source]¶

Return if self is formal.

A hyperplane arrangement is formal if it is 3-generated [Yuz1993], where \(k\)-generated is defined in minimal_generated_number().

EXAMPLES:

Sage

sage: P.<x,y,z> = HyperplaneArrangements(QQ)
sage: A = P(x, y, z, x+y+z, 2*x+y+z, 2*x+3*y+z, 2*x+3*y+4*z, 3*x+5*z, 3*x+4*y+5*z)
sage: B = P(x, y, z, x+y+z, 2*x+y+z, 2*x+3*y+z, 2*x+3*y+4*z, x+3*z, x+2*y+3*z)
sage: A.is_formal()
True
sage: B.is_formal()
False

Python

>>> from sage.all import *
>>> P = HyperplaneArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> A = P(x, y, z, x+y+z, Integer(2)*x+y+z, Integer(2)*x+Integer(3)*y+z, Integer(2)*x+Integer(3)*y+Integer(4)*z, Integer(3)*x+Integer(5)*z, Integer(3)*x+Integer(4)*y+Integer(5)*z)
>>> B = P(x, y, z, x+y+z, Integer(2)*x+y+z, Integer(2)*x+Integer(3)*y+z, Integer(2)*x+Integer(3)*y+Integer(4)*z, x+Integer(3)*z, x+Integer(2)*y+Integer(3)*z)
>>> A.is_formal()
True
>>> B.is_formal()
False

Sage Live

P.<x,y,z> = HyperplaneArrangements(QQ)
A = P(x, y, z, x+y+z, 2*x+y+z, 2*x+3*y+z, 2*x+3*y+4*z, 3*x+5*z, 3*x+4*y+5*z)
B = P(x, y, z, x+y+z, 2*x+y+z, 2*x+3*y+z, 2*x+3*y+4*z, x+3*z, x+2*y+3*z)
A.is_formal()
B.is_formal()

is_free(algorithm='singular')[source]¶

Return if self is free.

A hyperplane arrangement \(A\) is free if the module of derivations \(\operatorname{Der}(A)\) is a free \(S\)-module, where \(S\) is the corresponding symmetric space.

INPUT:

algorithm – (default: 'singular') can be one of the following:
- 'singular' – use Singular’s minimal free resolution
- 'BC' – use the algorithm given by Barakat and Cuntz in [BC2012] (much slower than using Singular)

ALGORITHM:

singular

Check that the minimal free resolution has length at most 2 by using Singular.

This implementation follows [BC2012] by constructing a chain of free modules

\[D(A) = D(A_n) < D(A_{n-1}) < \cdots < D(A_1) < D(A_0)\]

corresponding to some ordering of the arrangements \(A_0 \subset A_1 \subset \cdots \subset A_{n-1} \subset A_n = A\). Such a chain is found by using a backtracking algorithm.

EXAMPLES:

For type \(A\) arrangements, chordality is equivalent to freeness. We verify that in type \(A_3\):

Sage

sage: W = WeylGroup(['A', 3], prefix='s')                                   # needs sage.combinat sage.groups
sage: for x in W:                                                           # needs sage.combinat sage.groups
....:    A = x.inversion_arrangement()
....:    assert A.matroid().is_chordal() == A.is_free()

Python

>>> from sage.all import *
>>> W = WeylGroup(['A', Integer(3)], prefix='s')                                   # needs sage.combinat sage.groups
>>> for x in W:                                                           # needs sage.combinat sage.groups
...    A = x.inversion_arrangement()
...    assert A.matroid().is_chordal() == A.is_free()

Sage Live

W = WeylGroup(['A', 3], prefix='s')                                   # needs sage.combinat sage.groups
for x in W:                                                           # needs sage.combinat sage.groups
   A = x.inversion_arrangement()
   assert A.matroid().is_chordal() == A.is_free()

is_linear()[source]¶

Test whether all hyperplanes pass through the origin.

OUTPUT: boolean

EXAMPLES:

Sage

sage: a = hyperplane_arrangements.semiorder(3)
sage: a.is_linear()
False
sage: b = hyperplane_arrangements.braid(3)                                  # needs sage.graphs
sage: b.is_linear()                                                         # needs sage.graphs
True

sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: c = H(x+1, y+1)
sage: c.is_linear()
False
sage: c.is_central()
True

Python

>>> from sage.all import *
>>> a = hyperplane_arrangements.semiorder(Integer(3))
>>> a.is_linear()
False
>>> b = hyperplane_arrangements.braid(Integer(3))                                  # needs sage.graphs
>>> b.is_linear()                                                         # needs sage.graphs
True

>>> H = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> c = H(x+Integer(1), y+Integer(1))
>>> c.is_linear()
False
>>> c.is_central()
True

Sage Live

a = hyperplane_arrangements.semiorder(3)
a.is_linear()
b = hyperplane_arrangements.braid(3)                                  # needs sage.graphs
b.is_linear()                                                         # needs sage.graphs
H.<x,y> = HyperplaneArrangements(QQ)
c = H(x+1, y+1)
c.is_linear()
c.is_central()

is_separating_hyperplane(region1, region2, hyperplane)[source]¶

Test whether the hyperplane separates the given regions.

INPUT:

region1, region2 – polyhedra or list/tuple/iterable of coordinates which are regions of the arrangement or an interior point of a region
hyperplane – a hyperplane

OUTPUT: boolean; whether the hyperplane hyperplane separate the given regions

EXAMPLES:

Sage

sage: A.<x,y> = hyperplane_arrangements.coordinate(2)
sage: A.is_separating_hyperplane([1,1], [2,1], y)
False
sage: A.is_separating_hyperplane([1,1], [-1,1], x)
True
sage: r = A.region_containing_point([1,1])
sage: s = A.region_containing_point([-1,1])
sage: A.is_separating_hyperplane(r, s, x)
True

Python

>>> from sage.all import *
>>> A = hyperplane_arrangements.coordinate(Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
>>> A.is_separating_hyperplane([Integer(1),Integer(1)], [Integer(2),Integer(1)], y)
False
>>> A.is_separating_hyperplane([Integer(1),Integer(1)], [-Integer(1),Integer(1)], x)
True
>>> r = A.region_containing_point([Integer(1),Integer(1)])
>>> s = A.region_containing_point([-Integer(1),Integer(1)])
>>> A.is_separating_hyperplane(r, s, x)
True

Sage Live

A.<x,y> = hyperplane_arrangements.coordinate(2)
A.is_separating_hyperplane([1,1], [2,1], y)
A.is_separating_hyperplane([1,1], [-1,1], x)
r = A.region_containing_point([1,1])
s = A.region_containing_point([-1,1])
A.is_separating_hyperplane(r, s, x)

is_simplicial()[source]¶

Test whether the arrangement is simplicial.

A region is simplicial if the normal vectors of its bounding hyperplanes are linearly independent. A hyperplane arrangement is said to be simplicial if every region is simplicial.

OUTPUT: boolean; whether the hyperplane arrangement is simplicial

EXAMPLES:

Sage

sage: H.<x,y,z> = HyperplaneArrangements(QQ)
sage: A = H([[0,1,1,1], [0,1,2,3]])
sage: A.is_simplicial()
True
sage: A = H([[0,1,1,1], [0,1,2,3], [0,1,3,2]])
sage: A.is_simplicial()
True
sage: A = H([[0,1,1,1], [0,1,2,3], [0,1,3,2], [0,2,1,3]])
sage: A.is_simplicial()
False
sage: hyperplane_arrangements.braid(3).is_simplicial()                      # needs sage.graphs
True

Python

>>> from sage.all import *
>>> H = HyperplaneArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = H._first_ngens(3)
>>> A = H([[Integer(0),Integer(1),Integer(1),Integer(1)], [Integer(0),Integer(1),Integer(2),Integer(3)]])
>>> A.is_simplicial()
True
>>> A = H([[Integer(0),Integer(1),Integer(1),Integer(1)], [Integer(0),Integer(1),Integer(2),Integer(3)], [Integer(0),Integer(1),Integer(3),Integer(2)]])
>>> A.is_simplicial()
True
>>> A = H([[Integer(0),Integer(1),Integer(1),Integer(1)], [Integer(0),Integer(1),Integer(2),Integer(3)], [Integer(0),Integer(1),Integer(3),Integer(2)], [Integer(0),Integer(2),Integer(1),Integer(3)]])
>>> A.is_simplicial()
False
>>> hyperplane_arrangements.braid(Integer(3)).is_simplicial()                      # needs sage.graphs
True

Sage Live

H.<x,y,z> = HyperplaneArrangements(QQ)
A = H([[0,1,1,1], [0,1,2,3]])
A.is_simplicial()
A = H([[0,1,1,1], [0,1,2,3], [0,1,3,2]])
A.is_simplicial()
A = H([[0,1,1,1], [0,1,2,3], [0,1,3,2], [0,2,1,3]])
A.is_simplicial()
hyperplane_arrangements.braid(3).is_simplicial()                      # needs sage.graphs

matroid()[source]¶

Return the matroid associated to self.

Let \(A\) denote a central hyperplane arrangement and \(n_H\) the normal vector of some hyperplane \(H \in A\). We define a matroid \(M_A\) as the linear matroid spanned by \(\{ n_H | H \in A \}\). The matroid \(M_A\) is such that the lattice of flats of \(M\) is isomorphic to the intersection lattice of \(A\) (Proposition 3.6 in [Sta2007]).

EXAMPLES:

Sage

sage: P.<x,y,z> = HyperplaneArrangements(QQ)
sage: A = P(x, y, z, x+y+z, 2*x+y+z, 2*x+3*y+z, 2*x+3*y+4*z)
sage: M = A.matroid(); M
Linear matroid of rank 3 on 7 elements represented over the Rational Field

Python

>>> from sage.all import *
>>> P = HyperplaneArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> A = P(x, y, z, x+y+z, Integer(2)*x+y+z, Integer(2)*x+Integer(3)*y+z, Integer(2)*x+Integer(3)*y+Integer(4)*z)
>>> M = A.matroid(); M
Linear matroid of rank 3 on 7 elements represented over the Rational Field

Sage Live

P.<x,y,z> = HyperplaneArrangements(QQ)
A = P(x, y, z, x+y+z, 2*x+y+z, 2*x+3*y+z, 2*x+3*y+4*z)
M = A.matroid(); M

We check the lattice of flats is isomorphic to the intersection lattice:

Sage

sage: f = sum([list(M.flats(i)) for i in range(M.rank() + 1)], [])
sage: PF = Poset([f, lambda x, y: x < y])                                   # needs sage.combinat
sage: PF.is_isomorphic(A.intersection_poset())                              # needs sage.combinat
True

Python

>>> from sage.all import *
>>> f = sum([list(M.flats(i)) for i in range(M.rank() + Integer(1))], [])
>>> PF = Poset([f, lambda x, y: x < y])                                   # needs sage.combinat
>>> PF.is_isomorphic(A.intersection_poset())                              # needs sage.combinat
True

Sage Live

f = sum([list(M.flats(i)) for i in range(M.rank() + 1)], [])
PF = Poset([f, lambda x, y: x < y])                                   # needs sage.combinat
PF.is_isomorphic(A.intersection_poset())                              # needs sage.combinat

minimal_generated_number()[source]¶

Return the minimum \(k\) such that self is \(k\)-generated.

Let \(A\) be a central hyperplane arrangement. Let \(W_k\) denote the solution space of the linear system corresponding to the linear dependencies among the hyperplanes of \(A\) of length at most \(k\). We say \(A\) is \(k\)-generated if \(\dim W_k = \operatorname{rank} A\).

Equivalently this says all dependencies forming the Orlik-Terao ideal are generated by at most \(k\) hyperplanes.

EXAMPLES:

We construct Example 2.2 from [Yuz1993]:

Sage

sage: P.<x,y,z> = HyperplaneArrangements(QQ)
sage: A = P(x, y, z, x+y+z, 2*x+y+z, 2*x+3*y+z, 2*x+3*y+4*z, 3*x+5*z, 3*x+4*y+5*z)
sage: B = P(x, y, z, x+y+z, 2*x+y+z, 2*x+3*y+z, 2*x+3*y+4*z, x+3*z, x+2*y+3*z)
sage: A.minimal_generated_number()
3
sage: B.minimal_generated_number()
4

Python

>>> from sage.all import *
>>> P = HyperplaneArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> A = P(x, y, z, x+y+z, Integer(2)*x+y+z, Integer(2)*x+Integer(3)*y+z, Integer(2)*x+Integer(3)*y+Integer(4)*z, Integer(3)*x+Integer(5)*z, Integer(3)*x+Integer(4)*y+Integer(5)*z)
>>> B = P(x, y, z, x+y+z, Integer(2)*x+y+z, Integer(2)*x+Integer(3)*y+z, Integer(2)*x+Integer(3)*y+Integer(4)*z, x+Integer(3)*z, x+Integer(2)*y+Integer(3)*z)
>>> A.minimal_generated_number()
3
>>> B.minimal_generated_number()
4

Sage Live

P.<x,y,z> = HyperplaneArrangements(QQ)
A = P(x, y, z, x+y+z, 2*x+y+z, 2*x+3*y+z, 2*x+3*y+4*z, 3*x+5*z, 3*x+4*y+5*z)
B = P(x, y, z, x+y+z, 2*x+y+z, 2*x+3*y+z, 2*x+3*y+4*z, x+3*z, x+2*y+3*z)
A.minimal_generated_number()
B.minimal_generated_number()

n_bounded_regions()[source]¶

Return the number of (relatively) bounded regions.

OUTPUT:

An integer. The number of relatively bounded regions of the hyperplane arrangement.

EXAMPLES:

Sage

sage: A = hyperplane_arrangements.semiorder(3)
sage: A.n_bounded_regions()
7

Python

>>> from sage.all import *
>>> A = hyperplane_arrangements.semiorder(Integer(3))
>>> A.n_bounded_regions()
7

Sage Live

A = hyperplane_arrangements.semiorder(3)
A.n_bounded_regions()

n_hyperplanes()[source]¶

Return the number of hyperplanes in the arrangement.

OUTPUT: integer

EXAMPLES:

Sage

sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: A = H([1,1,0], [2,3,-1], [4,5,3])
sage: A.n_hyperplanes()
3
sage: len(A)    # equivalent
3

Python

>>> from sage.all import *
>>> H = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H([Integer(1),Integer(1),Integer(0)], [Integer(2),Integer(3),-Integer(1)], [Integer(4),Integer(5),Integer(3)])
>>> A.n_hyperplanes()
3
>>> len(A)    # equivalent
3

Sage Live

H.<x,y> = HyperplaneArrangements(QQ)
A = H([1,1,0], [2,3,-1], [4,5,3])
A.n_hyperplanes()
len(A)    # equivalent

n_regions()[source]¶

The number of regions of the hyperplane arrangement.

OUTPUT: integer

EXAMPLES:

Sage

sage: A = hyperplane_arrangements.semiorder(3)
sage: A.n_regions()
19

Python

>>> from sage.all import *
>>> A = hyperplane_arrangements.semiorder(Integer(3))
>>> A.n_regions()
19

Sage Live

A = hyperplane_arrangements.semiorder(3)
A.n_regions()

orlik_solomon_algebra(base_ring=None, ordering=None, **kwds)[source]¶

Return the Orlik-Solomon algebra of self.

INPUT:

base_ring – (default: the base field of self) the ring over which the Orlik-Solomon algebra will be defined
ordering – (optional) an ordering of the ground set

EXAMPLES:

Sage

sage: P.<x,y,z> = HyperplaneArrangements(QQ)
sage: A = P(x, y, z, x+y+z, 2*x+y+z, 2*x+3*y+z, 2*x+3*y+4*z)
sage: A.orlik_solomon_algebra()
Orlik-Solomon algebra of Linear matroid of rank 3 on 7 elements
 represented over the Rational Field
sage: A.orlik_solomon_algebra(base_ring=ZZ)
Orlik-Solomon algebra of Linear matroid of rank 3 on 7 elements
 represented over the Rational Field

Python

>>> from sage.all import *
>>> P = HyperplaneArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> A = P(x, y, z, x+y+z, Integer(2)*x+y+z, Integer(2)*x+Integer(3)*y+z, Integer(2)*x+Integer(3)*y+Integer(4)*z)
>>> A.orlik_solomon_algebra()
Orlik-Solomon algebra of Linear matroid of rank 3 on 7 elements
 represented over the Rational Field
>>> A.orlik_solomon_algebra(base_ring=ZZ)
Orlik-Solomon algebra of Linear matroid of rank 3 on 7 elements
 represented over the Rational Field

Sage Live

P.<x,y,z> = HyperplaneArrangements(QQ)
A = P(x, y, z, x+y+z, 2*x+y+z, 2*x+3*y+z, 2*x+3*y+4*z)
A.orlik_solomon_algebra()
A.orlik_solomon_algebra(base_ring=ZZ)

orlik_terao_algebra(base_ring=None, ordering=None, **kwds)[source]¶

Return the Orlik-Terao algebra of self.

INPUT:

base_ring – (default: the base field of self) the ring over which the Orlik-Terao algebra will be defined
ordering – (optional) an ordering of the ground set

EXAMPLES:

Sage

sage: P.<x,y,z> = HyperplaneArrangements(QQ)
sage: A = P(x, y, z, x+y+z, 2*x+y+z, 2*x+3*y+z, 2*x+3*y+4*z)
sage: A.orlik_terao_algebra()
Orlik-Terao algebra of Linear matroid of rank 3 on 7 elements
 represented over the Rational Field over Rational Field
sage: A.orlik_terao_algebra(base_ring=QQ['t'])
Orlik-Terao algebra of Linear matroid of rank 3 on 7 elements
 represented over the Rational Field
 over Univariate Polynomial Ring in t over Rational Field

Python

>>> from sage.all import *
>>> P = HyperplaneArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> A = P(x, y, z, x+y+z, Integer(2)*x+y+z, Integer(2)*x+Integer(3)*y+z, Integer(2)*x+Integer(3)*y+Integer(4)*z)
>>> A.orlik_terao_algebra()
Orlik-Terao algebra of Linear matroid of rank 3 on 7 elements
 represented over the Rational Field over Rational Field
>>> A.orlik_terao_algebra(base_ring=QQ['t'])
Orlik-Terao algebra of Linear matroid of rank 3 on 7 elements
 represented over the Rational Field
 over Univariate Polynomial Ring in t over Rational Field

Sage Live

P.<x,y,z> = HyperplaneArrangements(QQ)
A = P(x, y, z, x+y+z, 2*x+y+z, 2*x+3*y+z, 2*x+3*y+4*z)
A.orlik_terao_algebra()
A.orlik_terao_algebra(base_ring=QQ['t'])

plot(**kwds)[source]¶

Plot the hyperplane arrangement.

OUTPUT: a graphics object

EXAMPLES:

Sage

sage: L.<x, y> = HyperplaneArrangements(QQ)
sage: L(x, y, x+y-2).plot()                                                 # needs sage.plot
Graphics object consisting of 3 graphics primitives

Python

>>> from sage.all import *
>>> L = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> L(x, y, x+y-Integer(2)).plot()                                                 # needs sage.plot
Graphics object consisting of 3 graphics primitives

Sage Live

L.<x, y> = HyperplaneArrangements(QQ)
L(x, y, x+y-2).plot()                                                 # needs sage.plot

poincare_polynomial()[source]¶

Return the Poincaré polynomial of the hyperplane arrangement.

OUTPUT: the Poincaré polynomial in \(\QQ[x]\)

EXAMPLES:

Sage

sage: a = hyperplane_arrangements.coordinate(2)
sage: a.poincare_polynomial()
x^2 + 2*x + 1

Python

>>> from sage.all import *
>>> a = hyperplane_arrangements.coordinate(Integer(2))
>>> a.poincare_polynomial()
x^2 + 2*x + 1

Sage Live

a = hyperplane_arrangements.coordinate(2)
a.poincare_polynomial()

poset_of_regions(B=None, numbered_labels=True)[source]¶

Return the poset of regions for a central hyperplane arrangement.

The poset of regions is a partial order on the set of regions where the regions are ordered by \(R\leq R'\) if and only if \(S(R) \subseteq S(R')\) where \(S(R)\) is the set of hyperplanes which separate the region \(R\) from the base region \(B\).

INPUT:

B – a region (optional); if None, then an arbitrary region is chosen as the base region
numbered_labels – boolean (default: True); if True, then the elements of the poset are numbered. Else they are labelled with the regions themselves.

OUTPUT: a Poset object containing the poset of regions

EXAMPLES:

Sage

sage: H.<x,y,z> = HyperplaneArrangements(QQ)
sage: A = H([[0,1,1,1], [0,1,2,3]])
sage: A.poset_of_regions()                                                  # needs sage.combinat
Finite poset containing 4 elements

sage: # needs sage.combinat sage.graphs
sage: A = hyperplane_arrangements.braid(3)
sage: A.poset_of_regions()
Finite poset containing 6 elements
sage: A.poset_of_regions(numbered_labels=False)
Finite poset containing 6 elements
sage: A = hyperplane_arrangements.braid(4)
sage: A.poset_of_regions()
Finite poset containing 24 elements

sage: H.<x,y,z> = HyperplaneArrangements(QQ)
sage: A = H([[0,1,1,1], [0,1,2,3], [0,1,3,2], [0,2,1,3]])
sage: R = A.regions()
sage: base_region = R[3]
sage: A.poset_of_regions(B=base_region)                                     # needs sage.combinat
Finite poset containing 14 elements

Python

>>> from sage.all import *
>>> H = HyperplaneArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = H._first_ngens(3)
>>> A = H([[Integer(0),Integer(1),Integer(1),Integer(1)], [Integer(0),Integer(1),Integer(2),Integer(3)]])
>>> A.poset_of_regions()                                                  # needs sage.combinat
Finite poset containing 4 elements

>>> # needs sage.combinat sage.graphs
>>> A = hyperplane_arrangements.braid(Integer(3))
>>> A.poset_of_regions()
Finite poset containing 6 elements
>>> A.poset_of_regions(numbered_labels=False)
Finite poset containing 6 elements
>>> A = hyperplane_arrangements.braid(Integer(4))
>>> A.poset_of_regions()
Finite poset containing 24 elements

>>> H = HyperplaneArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = H._first_ngens(3)
>>> A = H([[Integer(0),Integer(1),Integer(1),Integer(1)], [Integer(0),Integer(1),Integer(2),Integer(3)], [Integer(0),Integer(1),Integer(3),Integer(2)], [Integer(0),Integer(2),Integer(1),Integer(3)]])
>>> R = A.regions()
>>> base_region = R[Integer(3)]
>>> A.poset_of_regions(B=base_region)                                     # needs sage.combinat
Finite poset containing 14 elements

Sage Live

H.<x,y,z> = HyperplaneArrangements(QQ)
A = H([[0,1,1,1], [0,1,2,3]])
A.poset_of_regions()                                                  # needs sage.combinat
# needs sage.combinat sage.graphs
A = hyperplane_arrangements.braid(3)
A.poset_of_regions()
A.poset_of_regions(numbered_labels=False)
A = hyperplane_arrangements.braid(4)
A.poset_of_regions()
H.<x,y,z> = HyperplaneArrangements(QQ)
A = H([[0,1,1,1], [0,1,2,3], [0,1,3,2], [0,2,1,3]])
R = A.regions()
base_region = R[3]
A.poset_of_regions(B=base_region)                                     # needs sage.combinat

primitive_eulerian_polynomial()[source]¶

Return the primitive Eulerian polynomial of self.

The primitive Eulerian polynomial of a hyperplane arrangement \(A\) is defined [BHS2023] by

\[P_A(z) := \sum_{X \in L} |\mu(B,X)| (z - 1)^{\mathrm{codim} X},\]

where \(L\) is the intersection poset of \(A\), \(B\) is the minimal element of \(L\) (here, the \(0\) dimensional subspace), and \(\mu\) is the Möbius function of \(L\).

OUTPUT: the primitive Eulerian polynomial in \(\ZZ[z]\)

EXAMPLES:

Sage

sage: A = hyperplane_arrangements.coordinate(2)
sage: A.primitive_eulerian_polynomial()                                     # needs sage.graphs
z^2
sage: B = hyperplane_arrangements.braid(3)
sage: B.primitive_eulerian_polynomial()                                     # needs sage.graphs
z^2 + z

sage: H = hyperplane_arrangements.Shi(['B',2]).cone()
sage: H.is_simplicial()
False
sage: H.primitive_eulerian_polynomial()                                     # needs sage.graphs
z^3 + 11*z^2 + 4*z

sage: H = hyperplane_arrangements.graphical(graphs.CycleGraph(4))
sage: H.primitive_eulerian_polynomial()                                     # needs sage.graphs
z^3 + 3*z^2 - z

Python

>>> from sage.all import *
>>> A = hyperplane_arrangements.coordinate(Integer(2))
>>> A.primitive_eulerian_polynomial()                                     # needs sage.graphs
z^2
>>> B = hyperplane_arrangements.braid(Integer(3))
>>> B.primitive_eulerian_polynomial()                                     # needs sage.graphs
z^2 + z

>>> H = hyperplane_arrangements.Shi(['B',Integer(2)]).cone()
>>> H.is_simplicial()
False
>>> H.primitive_eulerian_polynomial()                                     # needs sage.graphs
z^3 + 11*z^2 + 4*z

>>> H = hyperplane_arrangements.graphical(graphs.CycleGraph(Integer(4)))
>>> H.primitive_eulerian_polynomial()                                     # needs sage.graphs
z^3 + 3*z^2 - z

Sage Live

A = hyperplane_arrangements.coordinate(2)
A.primitive_eulerian_polynomial()                                     # needs sage.graphs
B = hyperplane_arrangements.braid(3)
B.primitive_eulerian_polynomial()                                     # needs sage.graphs
H = hyperplane_arrangements.Shi(['B',2]).cone()
H.is_simplicial()
H.primitive_eulerian_polynomial()                                     # needs sage.graphs
H = hyperplane_arrangements.graphical(graphs.CycleGraph(4))
H.primitive_eulerian_polynomial()                                     # needs sage.graphs

We verify Example 2.4 in [BHS2023] for \(k = 2,3,4,5\):

Sage

sage: R.<x,y> = HyperplaneArrangements(QQ)
sage: for k in range(2,6):                                                  # needs sage.graphs
....:     H = R([x+j*y for j in range(k)])
....:     H.primitive_eulerian_polynomial()
z^2
z^2 + z
z^2 + 2*z
z^2 + 3*z

Python

>>> from sage.all import *
>>> R = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = R._first_ngens(2)
>>> for k in range(Integer(2),Integer(6)):                                                  # needs sage.graphs
...     H = R([x+j*y for j in range(k)])
...     H.primitive_eulerian_polynomial()
z^2
z^2 + z
z^2 + 2*z
z^2 + 3*z

Sage Live

R.<x,y> = HyperplaneArrangements(QQ)
for k in range(2,6):                                                  # needs sage.graphs
    H = R([x+j*y for j in range(k)])
    H.primitive_eulerian_polynomial()

We verify Equation (4) in [BHS2023] on some examples:

Sage

sage: # needs sage.graphs
sage: R.<x> = ZZ[]
sage: Arr = [hyperplane_arrangements.braid(n) for n in range(2,6)]
sage: all(R(A.cocharacteristic_polynomial()(1/(x-1)) * (x-1)^A.dimension())
....:     == R(A.primitive_eulerian_polynomial()) for A in Arr)
True

Python

>>> from sage.all import *
>>> # needs sage.graphs
>>> R = ZZ['x']; (x,) = R._first_ngens(1)
>>> Arr = [hyperplane_arrangements.braid(n) for n in range(Integer(2),Integer(6))]
>>> all(R(A.cocharacteristic_polynomial()(Integer(1)/(x-Integer(1))) * (x-Integer(1))**A.dimension())
...     == R(A.primitive_eulerian_polynomial()) for A in Arr)
True

Sage Live

# needs sage.graphs
R.<x> = ZZ[]
Arr = [hyperplane_arrangements.braid(n) for n in range(2,6)]
all(R(A.cocharacteristic_polynomial()(1/(x-1)) * (x-1)^A.dimension())
    == R(A.primitive_eulerian_polynomial()) for A in Arr)

We compute types \(H_3\) and \(F_4\) in Table 1 of [BHS2023]:

Sage

sage: # needs sage.libs.gap
sage: W = CoxeterGroup(['H',3], implementation='matrix')
sage: A = HyperplaneArrangements(W.base_ring(), tuple(f'x{s}' for s in range(W.rank())))
sage: H = A([[0] + list(r) for r in W.positive_roots()])
sage: H.is_simplicial()                                                     # needs sage.graphs
True
sage: H.primitive_eulerian_polynomial()
z^3 + 28*z^2 + 16*z

sage: W = CoxeterGroup(['F',4], implementation='permutation')
sage: A = HyperplaneArrangements(QQ, tuple(f'x{s}' for s in range(W.rank())))
sage: H = A([[0] + list(r) for r in W.positive_roots()])
sage: H.primitive_eulerian_polynomial()     # long time                     # needs sage.graphs
z^4 + 116*z^3 + 220*z^2 + 48*z

Python

>>> from sage.all import *
>>> # needs sage.libs.gap
>>> W = CoxeterGroup(['H',Integer(3)], implementation='matrix')
>>> A = HyperplaneArrangements(W.base_ring(), tuple(f'x{s}' for s in range(W.rank())))
>>> H = A([[Integer(0)] + list(r) for r in W.positive_roots()])
>>> H.is_simplicial()                                                     # needs sage.graphs
True
>>> H.primitive_eulerian_polynomial()
z^3 + 28*z^2 + 16*z

>>> W = CoxeterGroup(['F',Integer(4)], implementation='permutation')
>>> A = HyperplaneArrangements(QQ, tuple(f'x{s}' for s in range(W.rank())))
>>> H = A([[Integer(0)] + list(r) for r in W.positive_roots()])
>>> H.primitive_eulerian_polynomial()     # long time                     # needs sage.graphs
z^4 + 116*z^3 + 220*z^2 + 48*z

Sage Live

# needs sage.libs.gap
W = CoxeterGroup(['H',3], implementation='matrix')
A = HyperplaneArrangements(W.base_ring(), tuple(f'x{s}' for s in range(W.rank())))
H = A([[0] + list(r) for r in W.positive_roots()])
H.is_simplicial()                                                     # needs sage.graphs
H.primitive_eulerian_polynomial()
W = CoxeterGroup(['F',4], implementation='permutation')
A = HyperplaneArrangements(QQ, tuple(f'x{s}' for s in range(W.rank())))
H = A([[0] + list(r) for r in W.positive_roots()])
H.primitive_eulerian_polynomial()     # long time                     # needs sage.graphs

We verify Proposition 2.5 in [BHS2023] on the braid arrangement \(B_k\) for \(k = 2,3,4,5\):

Sage

sage: B = [hyperplane_arrangements.braid(k) for k in range(2,6)]
sage: all(H.is_simplicial() for H in B)
True
sage: all(c > 0 for H in B                                                  # needs sage.graphs
....:     for c in H.primitive_eulerian_polynomial().coefficients())
True

Python

>>> from sage.all import *
>>> B = [hyperplane_arrangements.braid(k) for k in range(Integer(2),Integer(6))]
>>> all(H.is_simplicial() for H in B)
True
>>> all(c > Integer(0) for H in B                                                  # needs sage.graphs
...     for c in H.primitive_eulerian_polynomial().coefficients())
True

Sage Live

B = [hyperplane_arrangements.braid(k) for k in range(2,6)]
all(H.is_simplicial() for H in B)
all(c > 0 for H in B                                                  # needs sage.graphs
    for c in H.primitive_eulerian_polynomial().coefficients())

We verify Example 9.4 in [BHS2023] showing a hyperplane arrangement whose primitive Eulerian polynomial does not have real roots (in general, the graphical arrangement of a cycle graph corresponds to the arrangements in Example 9.4):

Sage

sage: # needs sage.graphs
sage: H = hyperplane_arrangements.graphical(graphs.CycleGraph(5))
sage: pep = H.primitive_eulerian_polynomial(); pep
z^4 + 6*z^3 - 4*z^2 + z
sage: pep.roots(QQbar)
[(-6.626418492719221?, 1),
 (0, 1),
 (0.3132092463596102? - 0.2298065541510677?*I, 1),
 (0.3132092463596102? + 0.2298065541510677?*I, 1)]
sage: pep.roots(AA)
[(-6.626418492719221?, 1), (0, 1)]

Python

>>> from sage.all import *
>>> # needs sage.graphs
>>> H = hyperplane_arrangements.graphical(graphs.CycleGraph(Integer(5)))
>>> pep = H.primitive_eulerian_polynomial(); pep
z^4 + 6*z^3 - 4*z^2 + z
>>> pep.roots(QQbar)
[(-6.626418492719221?, 1),
 (0, 1),
 (0.3132092463596102? - 0.2298065541510677?*I, 1),
 (0.3132092463596102? + 0.2298065541510677?*I, 1)]
>>> pep.roots(AA)
[(-6.626418492719221?, 1), (0, 1)]

Sage Live

# needs sage.graphs
H = hyperplane_arrangements.graphical(graphs.CycleGraph(5))
pep = H.primitive_eulerian_polynomial(); pep
pep.roots(QQbar)
pep.roots(AA)

rank()[source]¶

Return the rank.

OUTPUT:

The dimension of the span of the normals to the hyperplanes in the arrangement.

EXAMPLES:

Sage

sage: H.<x,y,z> = HyperplaneArrangements(QQ)
sage: A = H([[0, 1, 2, 3],[-3, 4, 5, 6]])
sage: A.dimension()
3
sage: A.rank()
2

sage: # needs sage.graphs
sage: B = hyperplane_arrangements.braid(3)
sage: B.hyperplanes()
(Hyperplane 0*t0 + t1 - t2 + 0,
 Hyperplane t0 - t1 + 0*t2 + 0,
 Hyperplane t0 + 0*t1 - t2 + 0)
sage: B.dimension()
3
sage: B.rank()
2

sage: p = polytopes.simplex(5, project=True)
sage: H = p.hyperplane_arrangement()
sage: H.rank()
5

Python

>>> from sage.all import *
>>> H = HyperplaneArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = H._first_ngens(3)
>>> A = H([[Integer(0), Integer(1), Integer(2), Integer(3)],[-Integer(3), Integer(4), Integer(5), Integer(6)]])
>>> A.dimension()
3
>>> A.rank()
2

>>> # needs sage.graphs
>>> B = hyperplane_arrangements.braid(Integer(3))
>>> B.hyperplanes()
(Hyperplane 0*t0 + t1 - t2 + 0,
 Hyperplane t0 - t1 + 0*t2 + 0,
 Hyperplane t0 + 0*t1 - t2 + 0)
>>> B.dimension()
3
>>> B.rank()
2

>>> p = polytopes.simplex(Integer(5), project=True)
>>> H = p.hyperplane_arrangement()
>>> H.rank()
5

Sage Live

H.<x,y,z> = HyperplaneArrangements(QQ)
A = H([[0, 1, 2, 3],[-3, 4, 5, 6]])
A.dimension()
A.rank()
# needs sage.graphs
B = hyperplane_arrangements.braid(3)
B.hyperplanes()
B.dimension()
B.rank()
p = polytopes.simplex(5, project=True)
H = p.hyperplane_arrangement()
H.rank()

region_containing_point(p)[source]¶

The region in the hyperplane arrangement containing a given point.

The base field must have characteristic zero.

INPUT:

p – point

OUTPUT:

A polyhedron. A ValueError is raised if the point is not interior to a region, that is, sits on a hyperplane.

EXAMPLES:

Sage

sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: A = H([(1,0), 0], [(0,1), 1], [(0,1), -1], [(1,-1), 0], [(1,1), 0])
sage: A.region_containing_point([1,2])
A 2-dimensional polyhedron in QQ^2 defined
as the convex hull of 2 vertices and 2 rays

Python

>>> from sage.all import *
>>> H = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H([(Integer(1),Integer(0)), Integer(0)], [(Integer(0),Integer(1)), Integer(1)], [(Integer(0),Integer(1)), -Integer(1)], [(Integer(1),-Integer(1)), Integer(0)], [(Integer(1),Integer(1)), Integer(0)])
>>> A.region_containing_point([Integer(1),Integer(2)])
A 2-dimensional polyhedron in QQ^2 defined
as the convex hull of 2 vertices and 2 rays

Sage Live

H.<x,y> = HyperplaneArrangements(QQ)
A = H([(1,0), 0], [(0,1), 1], [(0,1), -1], [(1,-1), 0], [(1,1), 0])
A.region_containing_point([1,2])

regions()[source]¶

Return the regions of the hyperplane arrangement.

The base field must have characteristic zero.

OUTPUT: a tuple containing the regions as polyhedra

The regions are the connected components of the complement of the union of the hyperplanes as a subset of \(\RR^n\).

EXAMPLES:

Sage

sage: a = hyperplane_arrangements.braid(2)                                  # needs sage.graphs
sage: a.regions()                                                           # needs sage.graphs
(A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex, 1 ray, 1 line,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex, 1 ray, 1 line)

sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: A = H(x, y+1)
sage: A.regions()
(A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex and 2 rays,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex and 2 rays,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex and 2 rays,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex and 2 rays)

sage: chessboard = []
sage: N = 8
sage: for x0 in range(N + 1):
....:     for y0 in range(N + 1):
....:         chessboard.extend([x-x0, y-y0])
sage: chessboard = H(chessboard)
sage: len(chessboard.bounded_regions())   # long time, 359 ms on a Core i7
64

Python

>>> from sage.all import *
>>> a = hyperplane_arrangements.braid(Integer(2))                                  # needs sage.graphs
>>> a.regions()                                                           # needs sage.graphs
(A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex, 1 ray, 1 line,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex, 1 ray, 1 line)

>>> H = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H(x, y+Integer(1))
>>> A.regions()
(A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex and 2 rays,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex and 2 rays,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex and 2 rays,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex and 2 rays)

>>> chessboard = []
>>> N = Integer(8)
>>> for x0 in range(N + Integer(1)):
...     for y0 in range(N + Integer(1)):
...         chessboard.extend([x-x0, y-y0])
>>> chessboard = H(chessboard)
>>> len(chessboard.bounded_regions())   # long time, 359 ms on a Core i7
64

Sage Live

a = hyperplane_arrangements.braid(2)                                  # needs sage.graphs
a.regions()                                                           # needs sage.graphs
H.<x,y> = HyperplaneArrangements(QQ)
A = H(x, y+1)
A.regions()
chessboard = []
N = 8
for x0 in range(N + 1):
    for y0 in range(N + 1):
        chessboard.extend([x-x0, y-y0])
chessboard = H(chessboard)
len(chessboard.bounded_regions())   # long time, 359 ms on a Core i7

Example 6 of [KP2020]:

Sage

sage: from itertools import product
sage: def zero_one(d):
....:     for x in product([0,1], repeat=d):
....:         if any(x):
....:             yield [0] + list(x)

sage: K.<x,y> = HyperplaneArrangements(QQ)
sage: A = K(*zero_one(2))
sage: len(A.regions())
6
sage: K.<x,y,z> = HyperplaneArrangements(QQ)
sage: A = K(*zero_one(3))
sage: len(A.regions())
32
sage: K.<x,y,z,w> = HyperplaneArrangements(QQ)
sage: A = K(*zero_one(4))
sage: len(A.regions())
370
sage: K.<x,y,z,w,r> = HyperplaneArrangements(QQ)
sage: A = K(*zero_one(5))
sage: len(A.regions())            # not tested (~25s)
11292

Python

>>> from sage.all import *
>>> from itertools import product
>>> def zero_one(d):
...     for x in product([Integer(0),Integer(1)], repeat=d):
...         if any(x):
...             yield [Integer(0)] + list(x)

>>> K = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = K._first_ngens(2)
>>> A = K(*zero_one(Integer(2)))
>>> len(A.regions())
6
>>> K = HyperplaneArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = K._first_ngens(3)
>>> A = K(*zero_one(Integer(3)))
>>> len(A.regions())
32
>>> K = HyperplaneArrangements(QQ, names=('x', 'y', 'z', 'w',)); (x, y, z, w,) = K._first_ngens(4)
>>> A = K(*zero_one(Integer(4)))
>>> len(A.regions())
370
>>> K = HyperplaneArrangements(QQ, names=('x', 'y', 'z', 'w', 'r',)); (x, y, z, w, r,) = K._first_ngens(5)
>>> A = K(*zero_one(Integer(5)))
>>> len(A.regions())            # not tested (~25s)
11292

Sage Live

from itertools import product
def zero_one(d):
    for x in product([0,1], repeat=d):
        if any(x):
            yield [0] + list(x)
K.<x,y> = HyperplaneArrangements(QQ)
A = K(*zero_one(2))
len(A.regions())
K.<x,y,z> = HyperplaneArrangements(QQ)
A = K(*zero_one(3))
len(A.regions())
K.<x,y,z,w> = HyperplaneArrangements(QQ)
A = K(*zero_one(4))
len(A.regions())
K.<x,y,z,w,r> = HyperplaneArrangements(QQ)
A = K(*zero_one(5))
len(A.regions())            # not tested (~25s)

It is possible to specify the backend:

Sage

sage: # needs sage.rings.number_field
sage: K.<q> = CyclotomicField(9)
sage: L.<r9> = NumberField((q + q**(-1)).minpoly(),
....:                      embedding=AA(q + q**-1))
sage: norms = [[1, 1/3*(-2*r9**2-r9+1), 0],
....:          [1, -r9**2 - r9, 0],
....:          [1, -r9**2 + 1, 0],
....:          [1, -r9**2, 0],
....:          [1, r9**2 - 4, -r9**2+3]]
sage: H.<x,y,z> = HyperplaneArrangements(L)
sage: A = H(backend='normaliz')
sage: for v in norms:
....:     a,b,c = v
....:     A = A.add_hyperplane(a*x + b*y + c*z)
sage: R = A.regions()                                       # optional - pynormaliz
sage: R[0].backend()                                        # optional - pynormaliz
'normaliz'

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = CyclotomicField(Integer(9), names=('q',)); (q,) = K._first_ngens(1)
>>> L = NumberField((q + q**(-Integer(1))).minpoly(),
...                      embedding=AA(q + q**-Integer(1)), names=('r9',)); (r9,) = L._first_ngens(1)
>>> norms = [[Integer(1), Integer(1)/Integer(3)*(-Integer(2)*r9**Integer(2)-r9+Integer(1)), Integer(0)],
...          [Integer(1), -r9**Integer(2) - r9, Integer(0)],
...          [Integer(1), -r9**Integer(2) + Integer(1), Integer(0)],
...          [Integer(1), -r9**Integer(2), Integer(0)],
...          [Integer(1), r9**Integer(2) - Integer(4), -r9**Integer(2)+Integer(3)]]
>>> H = HyperplaneArrangements(L, names=('x', 'y', 'z',)); (x, y, z,) = H._first_ngens(3)
>>> A = H(backend='normaliz')
>>> for v in norms:
...     a,b,c = v
...     A = A.add_hyperplane(a*x + b*y + c*z)
>>> R = A.regions()                                       # optional - pynormaliz
>>> R[Integer(0)].backend()                                        # optional - pynormaliz
'normaliz'

Sage Live

# needs sage.rings.number_field
K.<q> = CyclotomicField(9)
L.<r9> = NumberField((q + q**(-1)).minpoly(),
                     embedding=AA(q + q**-1))
norms = [[1, 1/3*(-2*r9**2-r9+1), 0],
         [1, -r9**2 - r9, 0],
         [1, -r9**2 + 1, 0],
         [1, -r9**2, 0],
         [1, r9**2 - 4, -r9**2+3]]
H.<x,y,z> = HyperplaneArrangements(L)
A = H(backend='normaliz')
for v in norms:
    a,b,c = v
    A = A.add_hyperplane(a*x + b*y + c*z)
R = A.regions()                                       # optional - pynormaliz
R[0].backend()                                        # optional - pynormaliz

restriction(hyperplane, repetitions=False)[source]¶

Return the restriction to a hyperplane.

INPUT:

hyperplane – a hyperplane of the hyperplane arrangement
repetitions – boolean (default: False); eliminate repetitions for ordered arrangements

OUTPUT:

The restriction \(\mathcal{A}_H\) of the hyperplane arrangement \(\mathcal{A}\) to the given hyperplane \(H\).

EXAMPLES:

Sage

sage: # needs sage.graphs
sage: A.<u,x,y,z> = hyperplane_arrangements.braid(4);  A
Arrangement of 6 hyperplanes of dimension 4 and rank 3
sage: H = A[0];  H
Hyperplane 0*u + 0*x + y - z + 0
sage: R = A.restriction(H); R
Arrangement <x - z | u - x | u - z>
sage: A.add_hyperplane(z).restriction(z)
Arrangement of 6 hyperplanes of dimension 3 and rank 3
sage: A.add_hyperplane(u).restriction(u)
Arrangement of 6 hyperplanes of dimension 3 and rank 3
sage: D = A.deletion(H);  D
Arrangement of 5 hyperplanes of dimension 4 and rank 3
sage: ca = A.characteristic_polynomial()
sage: cr = R.characteristic_polynomial()
sage: cd = D.characteristic_polynomial()
sage: ca
x^4 - 6*x^3 + 11*x^2 - 6*x
sage: cd - cr
x^4 - 6*x^3 + 11*x^2 - 6*x

Python

>>> from sage.all import *
>>> # needs sage.graphs
>>> A = hyperplane_arrangements.braid(Integer(4), names=('u', 'x', 'y', 'z',)); (u, x, y, z,) = A._first_ngens(4);  A
Arrangement of 6 hyperplanes of dimension 4 and rank 3
>>> H = A[Integer(0)];  H
Hyperplane 0*u + 0*x + y - z + 0
>>> R = A.restriction(H); R
Arrangement <x - z | u - x | u - z>
>>> A.add_hyperplane(z).restriction(z)
Arrangement of 6 hyperplanes of dimension 3 and rank 3
>>> A.add_hyperplane(u).restriction(u)
Arrangement of 6 hyperplanes of dimension 3 and rank 3
>>> D = A.deletion(H);  D
Arrangement of 5 hyperplanes of dimension 4 and rank 3
>>> ca = A.characteristic_polynomial()
>>> cr = R.characteristic_polynomial()
>>> cd = D.characteristic_polynomial()
>>> ca
x^4 - 6*x^3 + 11*x^2 - 6*x
>>> cd - cr
x^4 - 6*x^3 + 11*x^2 - 6*x

Sage Live

# needs sage.graphs
A.<u,x,y,z> = hyperplane_arrangements.braid(4);  A
H = A[0];  H
R = A.restriction(H); R
A.add_hyperplane(z).restriction(z)
A.add_hyperplane(u).restriction(u)
D = A.deletion(H);  D
ca = A.characteristic_polynomial()
cr = R.characteristic_polynomial()
cd = D.characteristic_polynomial()
ca
cd - cr

See also

deletion()

sign_vector(p)[source]¶

Indicates on which side of each hyperplane the given point \(p\) lies.

The base field must have characteristic zero.

INPUT:

p – point as a list/tuple/iterable

OUTPUT:

A vector whose entries are in \([-1, 0, +1]\).

EXAMPLES:

Sage

sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: A = H([(1,0), 0], [(0,1), 1]);  A
Arrangement <y + 1 | x>
sage: A.sign_vector([2, -2])
(-1, 1)
sage: A.sign_vector((-1, -1))
(0, -1)

Python

>>> from sage.all import *
>>> H = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H([(Integer(1),Integer(0)), Integer(0)], [(Integer(0),Integer(1)), Integer(1)]);  A
Arrangement <y + 1 | x>
>>> A.sign_vector([Integer(2), -Integer(2)])
(-1, 1)
>>> A.sign_vector((-Integer(1), -Integer(1)))
(0, -1)

Sage Live

H.<x,y> = HyperplaneArrangements(QQ)
A = H([(1,0), 0], [(0,1), 1]);  A
A.sign_vector([2, -2])
A.sign_vector((-1, -1))

unbounded_regions()[source]¶

Return the relatively bounded regions of the arrangement.

OUTPUT:

Tuple of polyhedra. The regions of the arrangement that are not relatively bounded. It is assumed that the arrangement is defined over the rationals.

See also

bounded_regions()

EXAMPLES:

Sage

sage: # needs sage.combinat
sage: A = hyperplane_arrangements.semiorder(3)
sage: B = A.essentialization()
sage: B.n_regions() - B.n_bounded_regions()
12
sage: B.unbounded_regions()
(A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 3 vertices and 1 ray,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 3 vertices and 1 ray,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex and 2 rays,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 3 vertices and 1 ray,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex and 2 rays,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 3 vertices and 1 ray,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex and 2 rays,
 A 2-dimensional polyhedron in QQ^2 defined
    as the convex hull of 3 vertices and 1 ray,
 A 2-dimensional polyhedron in QQ^2 defined
    as the convex hull of 1 vertex and 2 rays,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 3 vertices and 1 ray,
 A 2-dimensional polyhedron in QQ^2 defined
    as the convex hull of 1 vertex and 2 rays,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex and 2 rays)

Python

>>> from sage.all import *
>>> # needs sage.combinat
>>> A = hyperplane_arrangements.semiorder(Integer(3))
>>> B = A.essentialization()
>>> B.n_regions() - B.n_bounded_regions()
12
>>> B.unbounded_regions()
(A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 3 vertices and 1 ray,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 3 vertices and 1 ray,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex and 2 rays,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 3 vertices and 1 ray,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex and 2 rays,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 3 vertices and 1 ray,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex and 2 rays,
 A 2-dimensional polyhedron in QQ^2 defined
    as the convex hull of 3 vertices and 1 ray,
 A 2-dimensional polyhedron in QQ^2 defined
    as the convex hull of 1 vertex and 2 rays,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 3 vertices and 1 ray,
 A 2-dimensional polyhedron in QQ^2 defined
    as the convex hull of 1 vertex and 2 rays,
 A 2-dimensional polyhedron in QQ^2 defined
     as the convex hull of 1 vertex and 2 rays)

Sage Live

# needs sage.combinat
A = hyperplane_arrangements.semiorder(3)
B = A.essentialization()
B.n_regions() - B.n_bounded_regions()
B.unbounded_regions()

union(other)[source]¶

The union of self with other.

INPUT:

other – a hyperplane arrangement or something that can be converted into a hyperplane arrangement

OUTPUT: a new hyperplane arrangement

EXAMPLES:

Sage

sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: A = H([1,2,3], [0,1,1], [0,1,-1], [1,-1,0], [1,1,0])
sage: B = H([1,1,1], [1,-1,1], [1,0,-1])
sage: C = A.union(B); C
Arrangement of 8 hyperplanes of dimension 2 and rank 2
sage: C == A | B   # syntactic sugar
True

Python

>>> from sage.all import *
>>> H = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> A = H([Integer(1),Integer(2),Integer(3)], [Integer(0),Integer(1),Integer(1)], [Integer(0),Integer(1),-Integer(1)], [Integer(1),-Integer(1),Integer(0)], [Integer(1),Integer(1),Integer(0)])
>>> B = H([Integer(1),Integer(1),Integer(1)], [Integer(1),-Integer(1),Integer(1)], [Integer(1),Integer(0),-Integer(1)])
>>> C = A.union(B); C
Arrangement of 8 hyperplanes of dimension 2 and rank 2
>>> C == A | B   # syntactic sugar
True

Sage Live

H.<x,y> = HyperplaneArrangements(QQ)
A = H([1,2,3], [0,1,1], [0,1,-1], [1,-1,0], [1,1,0])
B = H([1,1,1], [1,-1,1], [1,0,-1])
C = A.union(B); C
C == A | B   # syntactic sugar

A single hyperplane is coerced into a hyperplane arrangement if necessary:

Sage

sage: A.union(x+y-1)
Arrangement of 6 hyperplanes of dimension 2 and rank 2
sage: A.add_hyperplane(x+y-1)    # alias
Arrangement of 6 hyperplanes of dimension 2 and rank 2

sage: P.<x,y> = HyperplaneArrangements(RR)
sage: C = P(2*x + 4*y + 5)
sage: C.union(A)
Arrangement of 6 hyperplanes of dimension 2 and rank 2

Python

>>> from sage.all import *
>>> A.union(x+y-Integer(1))
Arrangement of 6 hyperplanes of dimension 2 and rank 2
>>> A.add_hyperplane(x+y-Integer(1))    # alias
Arrangement of 6 hyperplanes of dimension 2 and rank 2

>>> P = HyperplaneArrangements(RR, names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> C = P(Integer(2)*x + Integer(4)*y + Integer(5))
>>> C.union(A)
Arrangement of 6 hyperplanes of dimension 2 and rank 2

Sage Live

A.union(x+y-1)
A.add_hyperplane(x+y-1)    # alias
P.<x,y> = HyperplaneArrangements(RR)
C = P(2*x + 4*y + 5)
C.union(A)

varchenko_matrix(names='h')[source]¶

Return the Varchenko matrix of the arrangement.

Let \(H_1, \ldots, H_s\) and \(R_1, \ldots, R_t\) denote the hyperplanes and regions, respectively, of the arrangement. Let \(S = \QQ[h_1, \ldots, h_s]\), a polynomial ring with indeterminate \(h_i\) corresponding to hyperplane \(H_i\). The Varchenko matrix is the \(t \times t\) matrix with \(i,j\)-th entry the product of those \(h_k\) such that \(H_k\) separates \(R_i\) and \(R_j\).

INPUT:

names – string or list/tuple/iterable of strings. The variable names for the polynomial ring \(S\)

OUTPUT: the Varchenko matrix

EXAMPLES:

Sage

sage: a = hyperplane_arrangements.coordinate(3)
sage: v = a.varchenko_matrix();  v
[       1       h2       h1    h1*h2 h0*h1*h2    h0*h1    h0*h2       h0]
[      h2        1    h1*h2       h1    h0*h1 h0*h1*h2       h0    h0*h2]
[      h1    h1*h2        1       h2    h0*h2       h0 h0*h1*h2    h0*h1]
[   h1*h2       h1       h2        1       h0    h0*h2    h0*h1 h0*h1*h2]
[h0*h1*h2    h0*h1    h0*h2       h0        1       h2       h1    h1*h2]
[   h0*h1 h0*h1*h2       h0    h0*h2       h2        1    h1*h2       h1]
[   h0*h2       h0 h0*h1*h2    h0*h1       h1    h1*h2        1       h2]
[      h0    h0*h2    h0*h1 h0*h1*h2    h1*h2       h1       h2        1]
sage: factor(det(v))
(h2 - 1)^4 * (h2 + 1)^4 * (h1 - 1)^4 * (h1 + 1)^4 * (h0 - 1)^4 * (h0 + 1)^4

Python

>>> from sage.all import *
>>> a = hyperplane_arrangements.coordinate(Integer(3))
>>> v = a.varchenko_matrix();  v
[       1       h2       h1    h1*h2 h0*h1*h2    h0*h1    h0*h2       h0]
[      h2        1    h1*h2       h1    h0*h1 h0*h1*h2       h0    h0*h2]
[      h1    h1*h2        1       h2    h0*h2       h0 h0*h1*h2    h0*h1]
[   h1*h2       h1       h2        1       h0    h0*h2    h0*h1 h0*h1*h2]
[h0*h1*h2    h0*h1    h0*h2       h0        1       h2       h1    h1*h2]
[   h0*h1 h0*h1*h2       h0    h0*h2       h2        1    h1*h2       h1]
[   h0*h2       h0 h0*h1*h2    h0*h1       h1    h1*h2        1       h2]
[      h0    h0*h2    h0*h1 h0*h1*h2    h1*h2       h1       h2        1]
>>> factor(det(v))
(h2 - 1)^4 * (h2 + 1)^4 * (h1 - 1)^4 * (h1 + 1)^4 * (h0 - 1)^4 * (h0 + 1)^4

Sage Live

a = hyperplane_arrangements.coordinate(3)
v = a.varchenko_matrix();  v
factor(det(v))

vertices(exclude_sandwiched=False)[source]¶

Return the vertices.

The vertices are the zero-dimensional faces, see face_vector().

INPUT:

exclude_sandwiched – boolean (default: False). Whether to exclude hyperplanes that are sandwiched between parallel hyperplanes. Useful if you only need the convex hull.

OUTPUT:

The vertices in a sorted tuple. Each vertex is returned as a vector in the ambient vector space.

EXAMPLES:

Sage

sage: # needs sage.combinat
sage: A = hyperplane_arrangements.Shi(3).essentialization()
sage: A.dimension()
2
sage: A.face_vector()
(6, 21, 16)
sage: A.vertices()
((-2/3, 1/3), (-1/3, -1/3), (0, -1), (0, 0), (1/3, -2/3), (2/3, -1/3))
sage: point2d(A.vertices(), size=20) + A.plot()                             # needs sage.plot
Graphics object consisting of 7 graphics primitives

sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: chessboard = []
sage: N = 8
sage: for x0 in range(N + 1):
....:     for y0 in range(N + 1):
....:         chessboard.extend([x-x0, y-y0])
sage: chessboard = H(chessboard)
sage: len(chessboard.vertices())
81
sage: chessboard.vertices(exclude_sandwiched=True)
((0, 0), (0, 8), (8, 0), (8, 8))

Python

>>> from sage.all import *
>>> # needs sage.combinat
>>> A = hyperplane_arrangements.Shi(Integer(3)).essentialization()
>>> A.dimension()
2
>>> A.face_vector()
(6, 21, 16)
>>> A.vertices()
((-2/3, 1/3), (-1/3, -1/3), (0, -1), (0, 0), (1/3, -2/3), (2/3, -1/3))
>>> point2d(A.vertices(), size=Integer(20)) + A.plot()                             # needs sage.plot
Graphics object consisting of 7 graphics primitives

>>> H = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> chessboard = []
>>> N = Integer(8)
>>> for x0 in range(N + Integer(1)):
...     for y0 in range(N + Integer(1)):
...         chessboard.extend([x-x0, y-y0])
>>> chessboard = H(chessboard)
>>> len(chessboard.vertices())
81
>>> chessboard.vertices(exclude_sandwiched=True)
((0, 0), (0, 8), (8, 0), (8, 8))

Sage Live

# needs sage.combinat
A = hyperplane_arrangements.Shi(3).essentialization()
A.dimension()
A.face_vector()
A.vertices()
point2d(A.vertices(), size=20) + A.plot()                             # needs sage.plot
H.<x,y> = HyperplaneArrangements(QQ)
chessboard = []
N = 8
for x0 in range(N + 1):
    for y0 in range(N + 1):
        chessboard.extend([x-x0, y-y0])
chessboard = H(chessboard)
len(chessboard.vertices())
chessboard.vertices(exclude_sandwiched=True)

whitney_data()[source]¶

Return the Whitney numbers.

OUTPUT:

A pair of integer matrices. The two matrices are the doubly-indexed Whitney numbers of the first or second kind, respectively. The \(i,j\)-th entry is the \(i,j\)-th doubly-indexed Whitney number.

EXAMPLES:

Sage

sage: # needs sage.combinat
sage: A = hyperplane_arrangements.Shi(3)
sage: A.whitney_data()
(
[  1  -6   9]  [ 1  6  6]
[  0   6 -15]  [ 0  6 15]
[  0   0   6], [ 0  0  6]
)

Python

>>> from sage.all import *
>>> # needs sage.combinat
>>> A = hyperplane_arrangements.Shi(Integer(3))
>>> A.whitney_data()
(
[  1  -6   9]  [ 1  6  6]
[  0   6 -15]  [ 0  6 15]
[  0   0   6], [ 0  0  6]
)

Sage Live

# needs sage.combinat
A = hyperplane_arrangements.Shi(3)
A.whitney_data()

whitney_number(k, kind=1)[source]¶

Return the k-th Whitney number.

If kind=1, this number is obtained by summing the Möbius function values \(mu(0, x)\) over all \(x\) in the intersection poset with \(\mathrm{rank}(x) = k\).

If kind=2, this number is the number of elements \(x, y\) in the intersection poset such that \(x \leq y\) with ranks \(i\) and \(j\), respectively.

See [GZ1983] for more details.

INPUT:

k – integer
kind – 1 or 2 (default: 1)

OUTPUT:

Integer. The k-th Whitney number.

EXAMPLES:

Sage

sage: # needs sage.combinat
sage: A = hyperplane_arrangements.Shi(3)
sage: A.whitney_number(0)
1
sage: A.whitney_number(1)
-6
sage: A.whitney_number(2)
9
sage: A.characteristic_polynomial()
x^3 - 6*x^2 + 9*x
sage: A.whitney_number(1, kind=2)
6
sage: p = A.intersection_poset()
sage: r = p.rank_function()
sage: len([i for i in p if r(i) == 1])
6

Python

>>> from sage.all import *
>>> # needs sage.combinat
>>> A = hyperplane_arrangements.Shi(Integer(3))
>>> A.whitney_number(Integer(0))
1
>>> A.whitney_number(Integer(1))
-6
>>> A.whitney_number(Integer(2))
9
>>> A.characteristic_polynomial()
x^3 - 6*x^2 + 9*x
>>> A.whitney_number(Integer(1), kind=Integer(2))
6
>>> p = A.intersection_poset()
>>> r = p.rank_function()
>>> len([i for i in p if r(i) == Integer(1)])
6

Sage Live

# needs sage.combinat
A = hyperplane_arrangements.Shi(3)
A.whitney_number(0)
A.whitney_number(1)
A.whitney_number(2)
A.characteristic_polynomial()
A.whitney_number(1, kind=2)
p = A.intersection_poset()
r = p.rank_function()
len([i for i in p if r(i) == 1])

class sage.geometry.hyperplane_arrangement.arrangement.HyperplaneArrangements(base_ring, names=())[source]¶

Bases: Parent, UniqueRepresentation

Hyperplane arrangements.

For more information on hyperplane arrangements, see sage.geometry.hyperplane_arrangement.arrangement.

INPUT:

base_ring – ring; the base ring
names – tuple of strings; the variable names

EXAMPLES:

Sage

sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: x
Hyperplane x + 0*y + 0
sage: x + y
Hyperplane x + y + 0
sage: H(x, y, x-1, y-1)
Arrangement <y - 1 | y | x - 1 | x>

Python

>>> from sage.all import *
>>> H = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = H._first_ngens(2)
>>> x
Hyperplane x + 0*y + 0
>>> x + y
Hyperplane x + y + 0
>>> H(x, y, x-Integer(1), y-Integer(1))
Arrangement <y - 1 | y | x - 1 | x>

Sage Live

H.<x,y> = HyperplaneArrangements(QQ)
x
x + y
H(x, y, x-1, y-1)

Element[source]¶: alias of HyperplaneArrangementElement

ambient_space()[source]¶

Return the ambient space.

The ambient space is the parent of hyperplanes. That is, new hyperplanes are always constructed internally from the ambient space instance.

EXAMPLES:

Sage

sage: L.<x, y> = HyperplaneArrangements(QQ)
sage: L.ambient_space()([(1,0), 0])
Hyperplane x + 0*y + 0
sage: L.ambient_space()([(1,0), 0]) == x
True

Python

>>> from sage.all import *
>>> L = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> L.ambient_space()([(Integer(1),Integer(0)), Integer(0)])
Hyperplane x + 0*y + 0
>>> L.ambient_space()([(Integer(1),Integer(0)), Integer(0)]) == x
True

Sage Live

L.<x, y> = HyperplaneArrangements(QQ)
L.ambient_space()([(1,0), 0])
L.ambient_space()([(1,0), 0]) == x

base_ring()[source]¶

Return the base ring.

OUTPUT: the base ring of the hyperplane arrangement

EXAMPLES:

Sage

sage: L.<x,y> = HyperplaneArrangements(QQ)
sage: L.base_ring()
Rational Field

Python

>>> from sage.all import *
>>> L = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> L.base_ring()
Rational Field

Sage Live

L.<x,y> = HyperplaneArrangements(QQ)
L.base_ring()

change_ring(base_ring)[source]¶

Return hyperplane arrangements over a different base ring.

INPUT:

base_ring – a ring; the new base ring

OUTPUT:

A new HyperplaneArrangements instance over the new base ring.

EXAMPLES:

Sage

sage: L.<x,y> = HyperplaneArrangements(QQ)
sage: L.gen(0)
Hyperplane x + 0*y + 0
sage: L.change_ring(RR).gen(0)
Hyperplane 1.00000000000000*x + 0.000000000000000*y + 0.000000000000000

Python

>>> from sage.all import *
>>> L = HyperplaneArrangements(QQ, names=('x', 'y',)); (x, y,) = L._first_ngens(2)
>>> L.gen(Integer(0))
Hyperplane x + 0*y + 0
>>> L.change_ring(RR).gen(Integer(0))
Hyperplane 1.00000000000000*x + 0.000000000000000*y + 0.000000000000000

Sage Live

L.<x,y> = HyperplaneArrangements(QQ)
L.gen(0)
L.change_ring(RR).gen(0)

gen(i)[source]¶

Return the \(i\)-th coordinate hyperplane.

INPUT:

i – integer

OUTPUT: a linear expression

EXAMPLES:

Sage

sage: L.<x, y, z> = HyperplaneArrangements(QQ);  L
Hyperplane arrangements in
 3-dimensional linear space over Rational Field with coordinates x, y, z
sage: L.gen(0)
Hyperplane x + 0*y + 0*z + 0

Python

>>> from sage.all import *
>>> L = HyperplaneArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = L._first_ngens(3);  L
Hyperplane arrangements in
 3-dimensional linear space over Rational Field with coordinates x, y, z
>>> L.gen(Integer(0))
Hyperplane x + 0*y + 0*z + 0

Sage Live

L.<x, y, z> = HyperplaneArrangements(QQ);  L
L.gen(0)

gens()[source]¶

Return the coordinate hyperplanes.

OUTPUT: a tuple of linear expressions, one for each linear variable

EXAMPLES:

Sage

sage: L = HyperplaneArrangements(QQ, ('x', 'y', 'z'))
sage: L.gens()
(Hyperplane x + 0*y + 0*z + 0,
 Hyperplane 0*x + y + 0*z + 0,
 Hyperplane 0*x + 0*y + z + 0)

Python

>>> from sage.all import *
>>> L = HyperplaneArrangements(QQ, ('x', 'y', 'z'))
>>> L.gens()
(Hyperplane x + 0*y + 0*z + 0,
 Hyperplane 0*x + y + 0*z + 0,
 Hyperplane 0*x + 0*y + z + 0)

Sage Live

L = HyperplaneArrangements(QQ, ('x', 'y', 'z'))
L.gens()

ngens()[source]¶

Return the number of linear variables.

OUTPUT: integer

EXAMPLES:

Sage

sage: L.<x, y, z> = HyperplaneArrangements(QQ);  L
Hyperplane arrangements in 3-dimensional linear space
 over Rational Field with coordinates x, y, z
sage: L.ngens()
3

Python

>>> from sage.all import *
>>> L = HyperplaneArrangements(QQ, names=('x', 'y', 'z',)); (x, y, z,) = L._first_ngens(3);  L
Hyperplane arrangements in 3-dimensional linear space
 over Rational Field with coordinates x, y, z
>>> L.ngens()
3

Sage Live

L.<x, y, z> = HyperplaneArrangements(QQ);  L
L.ngens()

Hyperplane Arrangements¶

Arrangements¶

Properties of Arrangements¶