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sage.geometry: Add some # optional, reformat doctests #35586

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sage.geometry: Add some # optional, reformat doctests #35586

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92709db
sage.geometry: Add some # optional, reformat doctests
mkoeppe Apr 30, 2023
cfb0331
sage.geometry: Add some # optional, reformat doctests (fixup)
mkoeppe Apr 30, 2023
76a4357
Merge remote-tracking branch 'upstream/develop' into sage_geometry_do…
mkoeppe May 21, 2023
3acd934
Merge remote-tracking branch 'upstream/develop' into sage_geometry_do…
mkoeppe May 23, 2023
28fcec5
sage.geometry: Fix # optional, reformat doctests
mkoeppe May 13, 2023
961f55a
src/sage/geometry/cone.py: Fix # optional
mkoeppe May 24, 2023
2be9942
src/sage/geometry/cone.py: Fix # optional
mkoeppe May 24, 2023
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440 changes: 220 additions & 220 deletions src/sage/geometry/cone.py
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95 changes: 51 additions & 44 deletions src/sage/geometry/fan.py
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Original file line number Diff line number Diff line change
@@ -1,4 +1,4 @@
# -*- coding: utf-8 -*-
# sage.doctest: optional - sage.graphs sage.combinat
r"""
Rational polyhedral fans

Expand Down Expand Up @@ -503,12 +503,12 @@ def Fan(cones, rays=None, lattice=None, check=True, normalize=True,
sage: fan = Fan([c1, c2], allow_arrangement=True)
sage: fan.ngenerating_cones()
7
sage: fan.plot() # optional - sage.plot
sage: fan.plot() # optional - sage.plot
Graphics3d Object

Cones of different dimension::

sage: c1 = Cone([(1,0),(0,1)])
sage: c1 = Cone([(1,0), (0,1)])
sage: c2 = Cone([(2,1)])
sage: c3 = Cone([(-1,-2)])
sage: fan = Fan([c1, c2, c3], allow_arrangement=True)
Expand All @@ -523,7 +523,7 @@ def Fan(cones, rays=None, lattice=None, check=True, normalize=True,
sage: c3 = Cone([[0, 1, 1], [1, 0, 1], [0, -1, 1], [-1, 0, 1]])
sage: c1 = Cone([[0, 0, 1]])
sage: fan1 = Fan([c1, c3], allow_arrangement=True)
sage: fan1.plot() # optional - sage.plot
sage: fan1.plot() # optional - sage.plot
Graphics3d Object

A 3-d cone and two 2-d cones::
Expand Down Expand Up @@ -1617,17 +1617,17 @@ def support_contains(self, *args):
True
sage: f.support_contains((-1,0))
False
sage: f.support_contains(f.lattice().dual()(1,0)) #random output (warning)
sage: f.support_contains(f.lattice().dual()(1,0)) # random output (warning)
False
sage: f.support_contains(f.lattice().dual()(1,0))
False
sage: f.support_contains(1)
False
sage: f.support_contains(0) # 0 converts to the origin in the lattice
True
sage: f.support_contains(1/2, sqrt(3))
sage: f.support_contains(1/2, sqrt(3)) # optional - sage.symbolic
True
sage: f.support_contains(-1/2, sqrt(3))
sage: f.support_contains(-1/2, sqrt(3)) # optional - sage.symbolic
False
"""
if len(args) == 1:
Expand Down Expand Up @@ -1668,10 +1668,10 @@ def cartesian_product(self, other, lattice=None):
EXAMPLES::

sage: K = ToricLattice(1, 'K')
sage: fan1 = Fan([[0],[1]],[(1,),(-1,)], lattice=K)
sage: fan1 = Fan([[0],[1]], [(1,),(-1,)], lattice=K)
sage: L = ToricLattice(2, 'L')
sage: fan2 = Fan(rays=[(1,0),(0,1),(-1,-1)],
....: cones=[[0,1],[1,2],[2,0]], lattice=L)
sage: fan2 = Fan(rays=[(1,0), (0,1), (-1,-1)],
....: cones=[[0,1], [1,2], [2,0]], lattice=L)
sage: fan1.cartesian_product(fan2)
Rational polyhedral fan in 3-d lattice K+L
sage: _.ngenerating_cones()
Expand Down Expand Up @@ -1733,14 +1733,14 @@ def common_refinement(self, other):

Refining a fan with itself gives itself::

sage: F0 = Fan2d([(1,0),(0,1),(-1,0),(0,-1)])
sage: F0 = Fan2d([(1,0), (0,1), (-1,0), (0,-1)])
sage: F0.common_refinement(F0) == F0
True

A more complex example with complete fans::

sage: F1 = Fan([[0],[1]],[(1,),(-1,)])
sage: F2 = Fan2d([(1,0),(1,1),(0,1),(-1,0),(0,-1)])
sage: F1 = Fan([[0],[1]], [(1,),(-1,)])
sage: F2 = Fan2d([(1,0), (1,1), (0,1), (-1,0), (0,-1)])
sage: F3 = F2.cartesian_product(F1)
sage: F4 = F1.cartesian_product(F2)
sage: FF = F3.common_refinement(F4)
Expand All @@ -1753,8 +1753,8 @@ def common_refinement(self, other):

An example with two non-complete fans with the same support::

sage: F5 = Fan2d([(1,0),(1,2),(0,1)])
sage: F6 = Fan2d([(1,0),(2,1),(0,1)])
sage: F5 = Fan2d([(1,0), (1,2), (0,1)])
sage: F6 = Fan2d([(1,0), (2,1), (0,1)])
sage: F5.common_refinement(F6).ngenerating_cones()
3

Expand Down Expand Up @@ -1997,7 +1997,7 @@ def cone_containing(self, *points):
TESTS::

sage: fan = Fan(cones=[(0,1,2,3), (0,1,4)],
....: rays=[(1,1,1), (1,-1,1), (1,-1,-1), (1,1,-1), (0,0,1)])
....: rays=[(1,1,1), (1,-1,1), (1,-1,-1), (1,1,-1), (0,0,1)])
sage: fan.cone_containing(0).rays()
N(1, 1, 1)
in 3-d lattice N
Expand Down Expand Up @@ -2525,18 +2525,18 @@ def vertex_graph(self):

EXAMPLES::

sage: dP8 = toric_varieties.dP8() # optional - palp sage.graphs
sage: g = dP8.fan().vertex_graph(); g # optional - palp sage.graphs
sage: dP8 = toric_varieties.dP8() # optional - palp sage.graphs
sage: g = dP8.fan().vertex_graph(); g # optional - palp sage.graphs
Graph on 4 vertices
sage: set(dP8.fan(1)) == set(g.vertices(sort=False)) # optional - palp sage.graphs
sage: set(dP8.fan(1)) == set(g.vertices(sort=False)) # optional - palp sage.graphs
True
sage: g.edge_labels() # all edge labels the same since every cone is smooth # optional - palp sage.graphs
[(1, 0), (1, 0), (1, 0), (1, 0)]

sage: g = toric_varieties.Cube_deformation(10).fan().vertex_graph() # optional - sage.graphs
sage: g.automorphism_group().order() # optional - sage.graphs sage.groups
sage: g = toric_varieties.Cube_deformation(10).fan().vertex_graph() # optional - sage.graphs
sage: g.automorphism_group().order() # optional - sage.graphs sage.groups
48
sage: g.automorphism_group(edge_labels=True).order() # optional - sage.graphs sage.groups
sage: g.automorphism_group(edge_labels=True).order() # optional - sage.graphs sage.groups
4
"""
from sage.geometry.cone import classify_cone_2d
Expand Down Expand Up @@ -2697,7 +2697,7 @@ def is_isomorphic(self, other) -> bool:
sage: rays = ((1, 1), (0, 1), (-1, -1), (1, 0))
sage: cones = [(0,1), (1,2), (2,3), (3,0)]
sage: fan1 = Fan(cones, rays)
sage: m = matrix([[-2,3],[1,-1]])
sage: m = matrix([[-2,3], [1,-1]])
sage: fan2 = Fan(cones, [vector(r)*m for r in rays])
sage: fan1.is_isomorphic(fan2)
True
Expand Down Expand Up @@ -2799,7 +2799,7 @@ def isomorphism(self, other):
sage: rays = ((1, 1), (0, 1), (-1, -1), (3, 1))
sage: cones = [(0,1), (1,2), (2,3), (3,0)]
sage: fan1 = Fan(cones, rays)
sage: m = matrix([[-2,3],[1,-1]])
sage: m = matrix([[-2,3], [1,-1]])
sage: fan2 = Fan(cones, [vector(r)*m for r in rays])

sage: fan1.isomorphism(fan2)
Expand Down Expand Up @@ -3190,7 +3190,8 @@ def primitive_collections(self):

EXAMPLES::

sage: fan = Fan([[0,1,3],[3,4],[2,0],[1,2,4]], [(-3, -2, 1), (0, 0, 1), (3, -2, 1), (-1, -1, 1), (1, -1, 1)])
sage: fan = Fan([[0,1,3], [3,4], [2,0], [1,2,4]],
....: [(-3, -2, 1), (0, 0, 1), (3, -2, 1), (-1, -1, 1), (1, -1, 1)])
sage: fan.primitive_collections()
[frozenset({0, 4}),
frozenset({2, 3}),
Expand Down Expand Up @@ -3250,9 +3251,11 @@ def Stanley_Reisner_ideal(self, ring):

EXAMPLES::

sage: fan = Fan([[0,1,3],[3,4],[2,0],[1,2,4]], [(-3, -2, 1), (0, 0, 1), (3, -2, 1), (-1, -1, 1), (1, -1, 1)])
sage: fan.Stanley_Reisner_ideal( PolynomialRing(QQ,5,'A, B, C, D, E') )
Ideal (A*E, C*D, A*B*C, B*D*E) of Multivariate Polynomial Ring in A, B, C, D, E over Rational Field
sage: fan = Fan([[0,1,3], [3,4], [2,0], [1,2,4]],
....: [(-3, -2, 1), (0, 0, 1), (3, -2, 1), (-1, -1, 1), (1, -1, 1)])
sage: fan.Stanley_Reisner_ideal(PolynomialRing(QQ, 5, 'A, B, C, D, E'))
Ideal (A*E, C*D, A*B*C, B*D*E) of
Multivariate Polynomial Ring in A, B, C, D, E over Rational Field
"""
generators_indices = self.primitive_collections()
return ring.ideal([prod([ring.gen(i) for i in sr])
Expand All @@ -3274,9 +3277,11 @@ def linear_equivalence_ideal(self, ring):

EXAMPLES::

sage: fan = Fan([[0,1,3],[3,4],[2,0],[1,2,4]], [(-3, -2, 1), (0, 0, 1), (3, -2, 1), (-1, -1, 1), (1, -1, 1)])
sage: fan.linear_equivalence_ideal( PolynomialRing(QQ,5,'A, B, C, D, E') )
Ideal (-3*A + 3*C - D + E, -2*A - 2*C - D - E, A + B + C + D + E) of Multivariate Polynomial Ring in A, B, C, D, E over Rational Field
sage: fan = Fan([[0,1,3],[3,4],[2,0],[1,2,4]],
....: [(-3, -2, 1), (0, 0, 1), (3, -2, 1), (-1, -1, 1), (1, -1, 1)])
sage: fan.linear_equivalence_ideal(PolynomialRing(QQ, 5, 'A, B, C, D, E'))
Ideal (-3*A + 3*C - D + E, -2*A - 2*C - D - E, A + B + C + D + E) of
Multivariate Polynomial Ring in A, B, C, D, E over Rational Field
"""
gens = []
for d in range(self.dim()):
Expand Down Expand Up @@ -3317,7 +3322,8 @@ def oriented_boundary(self, cone):
sage: fan = toric_varieties.P(3).fan() # optional - palp
sage: cone = fan(2)[0] # optional - palp
sage: bdry = fan.oriented_boundary(cone); bdry # optional - palp
-1-d cone of Rational polyhedral fan in 3-d lattice N + 1-d cone of Rational polyhedral fan in 3-d lattice N
-1-d cone of Rational polyhedral fan in 3-d lattice N
+ 1-d cone of Rational polyhedral fan in 3-d lattice N
sage: bdry[0] # optional - palp
(-1, 1-d cone of Rational polyhedral fan in 3-d lattice N)
sage: bdry[1] # optional - palp
Expand Down Expand Up @@ -3499,22 +3505,23 @@ def complex(self, base_ring=ZZ, extended=False):

EXAMPLES::

sage: fan = toric_varieties.P(3).fan() # optional - palp
sage: K_normal = fan.complex(); K_normal # optional - palp
sage: fan = toric_varieties.P(3).fan() # optional - palp
sage: K_normal = fan.complex(); K_normal # optional - palp
Chain complex with at most 4 nonzero terms over Integer Ring
sage: K_normal.homology() # optional - palp
sage: K_normal.homology() # optional - palp
{0: Z, 1: 0, 2: 0, 3: 0}
sage: K_extended = fan.complex(extended=True); K_extended # optional - palp
sage: K_extended = fan.complex(extended=True); K_extended # optional - palp
Chain complex with at most 5 nonzero terms over Integer Ring
sage: K_extended.homology() # optional - palp
sage: K_extended.homology() # optional - palp
{-1: 0, 0: 0, 1: 0, 2: 0, 3: 0}

Homology computations are much faster over `\QQ` if you do not
care about the torsion coefficients::

sage: toric_varieties.P2_123().fan().complex(extended=True, base_ring=QQ) # optional - palp
sage: toric_varieties.P2_123().fan().complex(extended=True, # optional - palp
....: base_ring=QQ)
Chain complex with at most 4 nonzero terms over Rational Field
sage: _.homology() # optional - palp
sage: _.homology() # optional - palp
{-1: Vector space of dimension 0 over Rational Field,
0: Vector space of dimension 0 over Rational Field,
1: Vector space of dimension 0 over Rational Field,
Expand Down Expand Up @@ -3542,15 +3549,15 @@ def complex(self, base_ring=ZZ, extended=False):
Things get more complicated for non-complete fans::

sage: fan = Fan([Cone([(1,1,1)]),
....: Cone([(1,0,0),(0,1,0)]),
....: Cone([(-1,0,0),(0,-1,0),(0,0,-1)])])
....: Cone([(1,0,0), (0,1,0)]),
....: Cone([(-1,0,0), (0,-1,0), (0,0,-1)])])
sage: fan.complex().homology()
{0: 0, 1: 0, 2: Z x Z, 3: 0}
sage: fan = Fan([Cone([(1,0,0),(0,1,0)]),
....: Cone([(-1,0,0),(0,-1,0),(0,0,-1)])])
sage: fan = Fan([Cone([(1,0,0), (0,1,0)]),
....: Cone([(-1,0,0), (0,-1,0), (0,0,-1)])])
sage: fan.complex().homology()
{0: 0, 1: 0, 2: Z, 3: 0}
sage: fan = Fan([Cone([(-1,0,0),(0,-1,0),(0,0,-1)])])
sage: fan = Fan([Cone([(-1,0,0), (0,-1,0), (0,0,-1)])])
sage: fan.complex().homology()
{0: 0, 1: 0, 2: 0, 3: 0}
"""
Expand Down
2 changes: 1 addition & 1 deletion src/sage/geometry/fan_isomorphism.py
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Original file line number Diff line number Diff line change
Expand Up @@ -216,7 +216,7 @@ def find_isomorphism(fan1, fan2, check=False):
sage: fan2 = Fan(cones, [vector(r)*m for r in rays])

sage: from sage.geometry.fan_isomorphism import find_isomorphism
sage: find_isomorphism(fan1, fan2, check=True)
sage: find_isomorphism(fan1, fan2, check=True) # optional - sage.graphs
Fan morphism defined by the matrix
[-2 3]
[ 1 -1]
Expand Down
15 changes: 8 additions & 7 deletions src/sage/geometry/fan_morphism.py
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Original file line number Diff line number Diff line change
@@ -1,3 +1,4 @@
# sage.doctest: optional - sage.graphs, sage.combinat
r"""
Morphisms between toric lattices compatible with fans

Expand Down Expand Up @@ -360,7 +361,7 @@ def _RISGIS(self):
sage: normal = NormalFan(diamond)
sage: N = face.lattice()
sage: fm = FanMorphism(identity_matrix(2),
....: normal, face, subdivide=True)
....: normal, face, subdivide=True)
sage: fm._RISGIS()
(frozenset({2}),
frozenset({3}),
Expand Down Expand Up @@ -774,7 +775,7 @@ def _support_error(self):
sage: quadrant = Fan([quadrant])
sage: quadrant_bl = quadrant.subdivide([(1,1)])
sage: fm = FanMorphism(identity_matrix(2),
....: quadrant, quadrant_bl, check=False)
....: quadrant, quadrant_bl, check=False)

Now we report that the morphism is invalid::

Expand Down Expand Up @@ -861,8 +862,8 @@ def _validate(self):

TESTS::

sage: trivialfan2 = Fan([],[],lattice=ToricLattice(2))
sage: trivialfan3 = Fan([],[],lattice=ToricLattice(3))
sage: trivialfan2 = Fan([], [], lattice=ToricLattice(2))
sage: trivialfan3 = Fan([], [], lattice=ToricLattice(3))
sage: FanMorphism(zero_matrix(2,3), trivialfan2, trivialfan3)
Fan morphism defined by the matrix
[0 0 0]
Expand Down Expand Up @@ -983,7 +984,7 @@ def image_cone(self, cone):
sage: normal = NormalFan(diamond)
sage: N = face.lattice()
sage: fm = FanMorphism(identity_matrix(2),
....: normal, face, subdivide=True)
....: normal, face, subdivide=True)
sage: fm.image_cone(Cone([(1,0)]))
1-d cone of Rational polyhedral fan in 2-d lattice N
sage: fm.image_cone(Cone([(1,1)]))
Expand Down Expand Up @@ -1855,8 +1856,8 @@ def factor(self):
sage: phi_i.codomain_fan() is phi.codomain_fan() # optional - palp
True

sage: trivialfan2 = Fan([],[],lattice=ToricLattice(2))
sage: trivialfan3 = Fan([],[],lattice=ToricLattice(3))
sage: trivialfan2 = Fan([], [], lattice=ToricLattice(2))
sage: trivialfan3 = Fan([], [], lattice=ToricLattice(3))
sage: f = FanMorphism(zero_matrix(2,3), trivialfan2, trivialfan3)
sage: [phi.matrix().dimensions() for phi in f.factor()]
[(0, 3), (0, 0), (2, 0)]
Expand Down
6 changes: 2 additions & 4 deletions src/sage/geometry/hasse_diagram.py
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Original file line number Diff line number Diff line change
Expand Up @@ -104,11 +104,9 @@ def lattice_from_incidences(atom_to_coatoms, coatom_to_atoms,
and we can compute the lattice as ::

sage: from sage.geometry.cone import lattice_from_incidences
sage: L = lattice_from_incidences(
....: atom_to_coatoms, coatom_to_atoms)
sage: L
sage: L = lattice_from_incidences(atom_to_coatoms, coatom_to_atoms); L # optional - sage.graphs
Finite lattice containing 8 elements with distinguished linear extension
sage: for level in L.level_sets(): print(level)
sage: for level in L.level_sets(): print(level) # optional - sage.graphs
[((), (0, 1, 2))]
[((0,), (0, 1)), ((1,), (0, 2)), ((2,), (1, 2))]
[((0, 1), (0,)), ((0, 2), (1,)), ((1, 2), (2,))]
Expand Down
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