sage.geometry: Modularization fixes (imports), # needs

src/sage/geometry/cone.py

@@ -206,17 +206,17 @@
 from copy import copy
 from warnings import warn
 
+from sage.misc.lazy_import import lazy_import
+lazy_import('sage.combinat.posets.posets', 'FinitePoset')
 from sage.arith.misc import GCD as gcd
 from sage.arith.functions import lcm
-from sage.combinat.posets.posets import FinitePoset
 from sage.geometry.point_collection import PointCollection
 from sage.geometry.polyhedron.constructor import Polyhedron
-from sage.geometry.hasse_diagram import lattice_from_incidences
+lazy_import('sage.geometry.hasse_diagram', 'lattice_from_incidences')
 from sage.geometry.toric_lattice import (ToricLattice, is_ToricLattice,
                                          is_ToricLatticeQuotient)
-from sage.geometry.toric_plotter import ToricPlotter, label_list
+lazy_import('sage.geometry.toric_plotter', ['ToricPlotter', 'label_list'])
 from sage.geometry.relative_interior import RelativeInterior
-from sage.graphs.digraph import DiGraph
 from sage.matrix.constructor import matrix
 from sage.matrix.matrix_space import MatrixSpace
 from sage.matrix.special import column_matrix
@@ -229,11 +229,10 @@
 from sage.rings.rational_field import QQ
 from sage.structure.all import SageObject, parent
 from sage.structure.richcmp import richcmp_method, richcmp
-from sage.geometry.integral_points import parallelotope_points
+lazy_import('sage.geometry.integral_points', 'parallelotope_points')
 from sage.geometry.convex_set import ConvexSet_closed
 import sage.geometry.abc
 
-from sage.misc.lazy_import import lazy_import
 from sage.features import PythonModule
 lazy_import('ppl', ['C_Polyhedron', 'Generator_System', 'Constraint_System',
                     'Linear_Expression', 'Poly_Con_Relation'],
@@ -627,10 +626,21 @@ def try_base_extend(ring):
     # If we don't have a lattice element, try successively
     # less-desirable ambient spaces until (as a last resort) we
     # attempt a numerical representation.
-    from sage.rings.qqbar import AA
-    from sage.rings.real_mpfr import RR
+    rings = [QQ]
+    try:
+        from sage.rings.qqbar import AA
+    except ImportError:
+        pass
+    else:
+        rings.append(AA)
+    try:
+        from sage.rings.real_mpfr import RR
+    except ImportError:
+        pass
+    else:
+        rings.append(RR)
 
-    for ring in [QQ, AA, RR]:
+    for ring in rings:
         p = try_base_extend(ring)
         if p is not None:
             return p
@@ -1103,7 +1113,7 @@ def plot(self, **options):
         EXAMPLES::
 
             sage: quadrant = Cone([(1,0), (0,1)])
-            sage: quadrant.plot()                                                       # needs sage.plot
+            sage: quadrant.plot()                                                       # needs sage.plot sage.symbolic
             Graphics object consisting of 9 graphics primitives
         """
         tp = ToricPlotter(options, self.lattice().degree(), self.rays())
@@ -2618,6 +2628,7 @@ def ConeFace(atoms, facets):
             else:
                 # Get face lattice as a sublattice of the ambient one
                 allowed_indices = frozenset(self._ambient_ray_indices)
+                from sage.graphs.digraph import DiGraph
                 L = DiGraph()
                 origin = \
                     self._ambient._face_lattice_function().bottom()
@@ -3270,11 +3281,12 @@ def is_isomorphic(self, other):
 
         We check that :issue:`18613` is fixed::
 
+            sage: # needs sage.graphs sage.groups
             sage: K = cones.trivial(0)
-            sage: K.is_isomorphic(K)                                                    # needs sage.graphs
+            sage: K.is_isomorphic(K)
             True
             sage: K = cones.trivial(1)
-            sage: K.is_isomorphic(K)                                                    # needs sage.graphs
+            sage: K.is_isomorphic(K)
             True
             sage: K = cones.trivial(2)
             sage: K.is_isomorphic(K)
@@ -3498,7 +3510,7 @@ def plot(self, **options):
         EXAMPLES::
 
             sage: quadrant = Cone([(1,0), (0,1)])
-            sage: quadrant.plot()                                                       # needs sage.plot
+            sage: quadrant.plot()                                                       # needs sage.plot sage.symbolic
             Graphics object consisting of 9 graphics primitives
         """
         # What to do with 3-d cones in 5-d? Use some projection method?

src/sage/geometry/fan.py

@@ -1,4 +1,4 @@
-# sage.doctest: optional - sage.graphs sage.combinat
+# sage.doctest: needs sage.graphs sage.combinat
 r"""
 Rational polyhedral fans
 
@@ -241,18 +241,18 @@
 import sage.geometry.abc
 
 from sage.structure.richcmp import richcmp_method, richcmp
-from sage.combinat.combination import Combinations
-from sage.combinat.posets.posets import FinitePoset
+from sage.misc.lazy_import import lazy_import
+lazy_import('sage.combinat.combination', 'Combinations')
+lazy_import('sage.combinat.posets.posets', 'FinitePoset')
 from sage.geometry.cone import (_ambient_space_point,
                                 Cone,
                                 ConvexRationalPolyhedralCone,
                                 IntegralRayCollection,
                                 normalize_rays)
-from sage.geometry.hasse_diagram import lattice_from_incidences
+lazy_import('sage.geometry.hasse_diagram', 'lattice_from_incidences')
 from sage.geometry.point_collection import PointCollection
 from sage.geometry.toric_lattice import ToricLattice, is_ToricLattice
-from sage.geometry.toric_plotter import ToricPlotter
-from sage.graphs.digraph import DiGraph
+lazy_import('sage.geometry.toric_plotter', 'ToricPlotter')
 from sage.matrix.constructor import matrix
 from sage.misc.cachefunc import cached_method
 from sage.misc.timing import walltime
@@ -1397,7 +1397,7 @@ def _compute_cone_lattice(self):
         not even need to check if it is complete::
 
             sage: fan = toric_varieties.P1xP1().fan()                                   # needs palp
-            sage: fan.cone_lattice() # indirect doctest                                 # needs palp
+            sage: fan.cone_lattice()  # indirect doctest                                # needs palp
             Finite lattice containing 10 elements with distinguished linear extension
 
         These 10 elements are: 1 origin, 4 rays, 4 generating cones, 1 fan.
@@ -1467,6 +1467,7 @@ def FanFace(rays, cones):
         else:
             # For general fans we will "merge" face lattices of generating
             # cones.
+            from sage.graphs.digraph import DiGraph
             L = DiGraph()
             face_to_rays = {}  # face |---> (indices of fan rays)
             rays_to_index = {}  # (indices of fan rays) |---> face index

src/sage/geometry/fan_morphism.py

@@ -1,4 +1,4 @@
-# sage.doctest: optional - sage.graphs, sage.combinat
+# sage.doctest: needs sage.combinat sage.graphs
 r"""
 Morphisms between toric lattices compatible with fans
 
@@ -85,8 +85,6 @@
 from sage.matrix.special import identity_matrix
 from sage.structure.element import is_Matrix
 from sage.misc.cachefunc import cached_method
-from sage.misc.latex import latex
-from sage.misc.misc import walltime
 from sage.misc.misc_c import prod
 from sage.modules.free_module_morphism import FreeModuleMorphism
 from sage.rings.infinity import Infinity
@@ -505,6 +503,7 @@ def _latex_(self):
             0 & 1
             \end{array}\right) : \Sigma^{2} \to \Sigma^{2}
         """
+        from sage.misc.latex import latex
         return (r"%s : %s \to %s" % (latex(self.matrix()),
                         latex(self.domain_fan()), latex(self.codomain_fan())))
 

src/sage/geometry/hyperplane_arrangement/arrangement.py

@@ -1009,10 +1009,10 @@ def cocharacteristic_polynomial(self):
         EXAMPLES::
 
             sage: A = hyperplane_arrangements.coordinate(2)
-            sage: A.cocharacteristic_polynomial()
+            sage: A.cocharacteristic_polynomial()                                       # needs sage.graphs
             z^2 + 2*z + 1
             sage: B = hyperplane_arrangements.braid(3)
-            sage: B.cocharacteristic_polynomial()
+            sage: B.cocharacteristic_polynomial()                                       # needs sage.graphs
             2*z^3 + 3*z^2 + z
 
         TESTS::
@@ -1059,26 +1059,26 @@ def primitive_eulerian_polynomial(self):
         EXAMPLES::
 
             sage: A = hyperplane_arrangements.coordinate(2)
-            sage: A.primitive_eulerian_polynomial()
+            sage: A.primitive_eulerian_polynomial()                                     # needs sage.graphs
             z^2
             sage: B = hyperplane_arrangements.braid(3)
-            sage: B.primitive_eulerian_polynomial()
+            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()
+            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()
+            sage: H.primitive_eulerian_polynomial()                                     # needs sage.graphs
             z^3 + 3*z^2 - z
 
         We verify Example 2.4 in [BHS2023]_ for `k = 2,3,4,5`::
 
             sage: R.<x,y> = HyperplaneArrangements(QQ)
-            sage: for k in range(2,6):
+            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
@@ -1088,6 +1088,7 @@ def primitive_eulerian_polynomial(self):
 
         We verify Equation (4) in [BHS2023]_ on some examples::
 
+            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())
@@ -1096,18 +1097,19 @@ def primitive_eulerian_polynomial(self):
 
         We compute types `H_3` and `F_4` in Table 1 of [BHS2023]_::
 
+            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()
+            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
+            sage: H.primitive_eulerian_polynomial()     # long time                     # needs sage.graphs
             z^4 + 116*z^3 + 220*z^2 + 48*z
 
         We verify Proposition 2.5 in [BHS2023]_ on the braid arrangement
@@ -1116,14 +1118,16 @@ def primitive_eulerian_polynomial(self):
             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 for c in H.primitive_eulerian_polynomial().coefficients())
+            sage: all(c > 0 for H in B                                                  # needs sage.graphs
+            ....:     for c in H.primitive_eulerian_polynomial().coefficients())
             True
 
         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: # 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

src/sage/geometry/hyperplane_arrangement/check_freeness.py

@@ -1,4 +1,4 @@
-# sage.doctest: optional - sage.libs.singular
+# sage.doctest: needs sage.libs.singular
 r"""
 Helper Functions For Freeness Of Hyperplane Arrangements
 

src/sage/geometry/hyperplane_arrangement/library.py

@@ -12,20 +12,14 @@
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 
-from sage.graphs.graph_generators import graphs
+from sage.arith.misc import binomial
+from sage.geometry.hyperplane_arrangement.arrangement import HyperplaneArrangements
 from sage.matrix.constructor import matrix, random_matrix
+from sage.misc.misc_c import prod
 from sage.rings.integer_ring import ZZ
+from sage.rings.polynomial.polynomial_ring import polygen
 from sage.rings.rational_field import QQ
 from sage.rings.semirings.non_negative_integer_semiring import NN
-from sage.misc.misc_c import prod
-
-from sage.combinat.combinat import stirling_number2
-from sage.combinat.root_system.cartan_type import CartanType
-from sage.combinat.root_system.root_system import RootSystem
-from sage.arith.misc import binomial
-from sage.rings.polynomial.polynomial_ring import polygen
-
-from sage.geometry.hyperplane_arrangement.arrangement import HyperplaneArrangements
 
 
 def make_parent(base_ring, dimension, names=None):
@@ -93,9 +87,11 @@ def braid(self, n, K=QQ, names=None):
             sage: hyperplane_arrangements.braid(4)                                      # needs sage.graphs
             Arrangement of 6 hyperplanes of dimension 4 and rank 3
         """
+        from sage.graphs.graph_generators import graphs
+
         x = polygen(QQ, 'x')
         A = self.graphical(graphs.CompleteGraph(n), K, names=names)
-        charpoly = prod(x-i for i in range(n))
+        charpoly = prod(x - i for i in range(n))
         A.characteristic_polynomial.set_cache(charpoly)
         return A
 
@@ -278,10 +274,11 @@ def Coxeter(self, data, K=QQ, names=None):
         If the Cartan type is not crystallographic, the Coxeter arrangement
         is not implemented yet::
 
-            sage: hyperplane_arrangements.Coxeter("H3")
+            sage: hyperplane_arrangements.Coxeter("H3")                                 # needs sage.libs.gap
             Traceback (most recent call last):
             ...
-            NotImplementedError: Coxeter arrangements are not implemented for non crystallographic Cartan types
+            NotImplementedError: Coxeter arrangements are not implemented
+            for non crystallographic Cartan types
 
         The characteristic polynomial is pre-computed using the results
         of Terao, see [Ath2000]_::
@@ -290,7 +287,10 @@ def Coxeter(self, data, K=QQ, names=None):
             sage: hyperplane_arrangements.Coxeter("A3").characteristic_polynomial()
             x^3 - 6*x^2 + 11*x - 6
         """
+        from sage.combinat.root_system.cartan_type import CartanType
+        from sage.combinat.root_system.root_system import RootSystem
         from sage.combinat.root_system.weyl_group import WeylGroup
+
         if data in NN:
             cartan_type = CartanType(["A", data - 1])
         else:
@@ -492,6 +492,8 @@ def Ish(self, n, K=QQ, names=None):
 
         - [AR2012]_
         """
+        from sage.combinat.combinat import stirling_number2
+
         H = make_parent(K, n, names)
         x = H.gens()
         hyperplanes = []
@@ -541,7 +543,7 @@ def IshB(self, n, K=QQ, names=None):
              Hyperplane t0 + 0*t1 + 0,
              Hyperplane t0 + 0*t1 + 1,
              Hyperplane t0 + t1 + 0)
-            sage: a.cone().is_free()
+            sage: a.cone().is_free()                                                    # needs sage.libs.singular
             True
 
         .. PLOT::
@@ -664,6 +666,8 @@ def semiorder(self, n, K=QQ, names=None):
             sage: h.characteristic_polynomial()         # long time
             x^5 - 20*x^4 + 180*x^3 - 790*x^2 + 1380*x
         """
+        from sage.combinat.combinat import stirling_number2
+
         H = make_parent(K, n, names)
         x = H.gens()
         hyperplanes = []
@@ -778,6 +782,9 @@ def Shi(self, data, K=QQ, names=None, m=1):
             sage: h.characteristic_polynomial()
             x^3 - 54*x^2 + 972*x - 5832
         """
+        from sage.combinat.root_system.cartan_type import CartanType
+        from sage.combinat.root_system.root_system import RootSystem
+
         if data in NN:
             cartan_type = CartanType(["A", data - 1])
         else:

src/sage/geometry/hyperplane_arrangement/plot.py

@@ -115,7 +115,6 @@
 lazy_import("sage.plot.text", "text")
 lazy_import("sage.plot.point", "point")
 lazy_import("sage.plot.plot", "parametric_plot")
-from sage.symbolic.ring import SR
 
 
 def plot(hyperplane_arrangement, **kwds):
@@ -423,6 +422,7 @@ def plot_hyperplane(hyperplane, **kwds):
     elif hyperplane.dimension() == 1: # a line in the plane
         pnt = hyperplane.point()
         w = hyperplane.linear_part().matrix()
+        from sage.symbolic.ring import SR
         t = SR.var('t')
         if ranges_set:
             if isinstance(ranges, (list, tuple)):
@@ -443,6 +443,7 @@ def plot_hyperplane(hyperplane, **kwds):
     elif hyperplane.dimension() == 2: # a plane in 3-space
         pnt = hyperplane.point()
         w = hyperplane.linear_part().matrix()
+        from sage.symbolic.ring import SR
         s, t = SR.var('s t')
         if ranges_set:
             if isinstance(ranges, (list, tuple)):