Trac #34195: sage.geometry.polyhedron: More # optional - sage.rings.number_field

We also remove some module-level imports of `AA` etc.

Split out from #32432.

URL: https://trac.sagemath.org/34195
Reported by: mkoeppe
Ticket author(s): Matthias Koeppe
Reviewer(s): Jonathan Kliem

Author: Release Manager

src/sage/geometry/convex_set.py

@@ -873,10 +873,14 @@ def _test_contains(self, tester=None, **options):
                 tester.assertEqual(contains_space_point, self.contains(ambient_point))
             tester.assertEqual(contains_space_point, self.contains(space_coords))
             if space.base_ring().is_exact():
-                from sage.rings.qqbar import AA
-                ext_space = self.ambient_vector_space(AA)
-                ext_space_point = ext_space(space_point)
-                tester.assertEqual(contains_space_point, self.contains(ext_space_point))
+                try:
+                    from sage.rings.qqbar import AA
+                except ImportError:
+                    pass
+                else:
+                    ext_space = self.ambient_vector_space(AA)
+                    ext_space_point = ext_space(space_point)
+                    tester.assertEqual(contains_space_point, self.contains(ext_space_point))
             try:
                 from sage.symbolic.ring import SR
                 symbolic_space = self.ambient_vector_space(SR)

src/sage/geometry/polyhedron/backend_normaliz.py

@@ -30,8 +30,8 @@
 lazy_import('PyNormaliz', ['NmzResult', 'NmzCompute', 'NmzCone', 'NmzConeCopy'],
             feature=sage.features.normaliz.PyNormaliz())
 
-from sage.rings.all import ZZ, QQ, QQbar
-from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
+from sage.rings.integer_ring import ZZ
+from sage.rings.rational_field import QQ
 from sage.arith.functions import LCM_list
 from sage.misc.functional import denominator
 from sage.matrix.constructor import vector
@@ -1082,10 +1082,11 @@ def _number_field_triple(normaliz_field):
             sage: Pn._number_field_triple(QuadraticField(5))  # optional - sage.rings.number_field
             ['a^2 - 5', 'a', '[2.236067977499789 +/- 8.06e-16]']
         """
-        from sage.rings.real_arb import RealBallField
         R = normaliz_field
         if R is QQ:
             return None
+        from sage.rings.real_arb import RealBallField
+        from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
         emb = RealBallField(53)(R.gen(0))
         gen = 'a'
         R_a = PolynomialRing(QQ, gen)
@@ -2336,8 +2337,9 @@ class functions.
             ((t^4 + 3*t^3 + 8*t^2 + 3*t + 1)/(t + 1), (3*t^3 + 2*t^2 + 3*t)/(t + 1))
         """
         from sage.groups.conjugacy_classes import ConjugacyClassGAP
-        from sage.rings.all import CyclotomicField
-        from sage.matrix.all import MatrixSpace
+        from sage.rings.number_field.number_field import CyclotomicField
+        from sage.rings.qqbar import QQbar
+        from sage.matrix.matrix_space import MatrixSpace
         from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
         from sage.matrix.special import identity_matrix
         # Setting the group
@@ -2475,6 +2477,8 @@ def _Hstar_as_rat_fct(self, initial_Hstar):
             [ 1  1 -1 -1  1]
             [ 2  0  0  0 -2]
         """
+        from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
+        from sage.rings.qqbar import QQbar
         chi_vars = ','.join('chi_{}'.format(i) for i in range(len(initial_Hstar)))
         Chi_ring = PolynomialRing(QQbar, chi_vars)
         virtual_ring = PolynomialRing(Chi_ring, initial_Hstar.base_ring().gens())

src/sage/geometry/polyhedron/base0.py

@@ -1366,11 +1366,9 @@ def cdd_Hrepresentation(self):
         except AttributeError:
             from sage.rings.integer_ring import ZZ
             from sage.rings.rational_field import QQ
-            from sage.rings.real_double import RDF
-
             if self.base_ring() is ZZ or self.base_ring() is QQ:
                 cdd_type = 'rational'
-            elif self.base_ring() is RDF:
+            elif isinstance(self.base_ring(), sage.rings.abc.RealDoubleField):
                 cdd_type = 'real'
             else:
                 raise TypeError('the base ring must be ZZ, QQ, or RDF')
@@ -1431,11 +1429,9 @@ def cdd_Vrepresentation(self):
         except AttributeError:
             from sage.rings.integer_ring import ZZ
             from sage.rings.rational_field import QQ
-            from sage.rings.real_double import RDF
-
             if self.base_ring() is ZZ or self.base_ring() is QQ:
                 cdd_type = 'rational'
-            elif self.base_ring() is RDF:
+            elif isinstance(self.base_ring(), sage.rings.abc.RealDoubleField):
                 cdd_type = 'real'
             else:
                 raise TypeError('the base ring must be ZZ, QQ, or RDF')

src/sage/geometry/polyhedron/base3.py

@@ -135,8 +135,8 @@ def slack_matrix(self):
             [1 0 1 0 0 1]
             [1 0 0 0 1 1]
 
-            sage: P = polytopes.dodecahedron().faces(2)[0].as_polyhedron()
-            sage: P.slack_matrix()
+            sage: P = polytopes.dodecahedron().faces(2)[0].as_polyhedron()                          # optional - sage.rings.number_field
+            sage: P.slack_matrix()                                                                  # optional - sage.rings.number_field
             [1/2*sqrt5 - 1/2               0               0               1 1/2*sqrt5 - 1/2               0]
             [              0               0 1/2*sqrt5 - 1/2 1/2*sqrt5 - 1/2               1               0]
             [              0 1/2*sqrt5 - 1/2               1               0 1/2*sqrt5 - 1/2               0]
@@ -1870,6 +1870,7 @@ def _test_combinatorial_face_as_combinatorial_polyhedron(self, tester=None, **op
             D2._test_bitsets(tester, **options)
             try:
                 import sage.graphs.graph
+                assert sage.graphs.graph  # to muffle pyflakes
             except ImportError:
                 pass
             else:

src/sage/geometry/polyhedron/base6.py

@@ -34,7 +34,6 @@
 from sage.modules.vector_space_morphism import linear_transformation
 from sage.matrix.constructor import matrix
 from sage.modules.free_module_element import vector
-from sage.rings.qqbar import AA
 from sage.geometry.convex_set import AffineHullProjectionData
 from .base5 import Polyhedron_base5
 
@@ -741,6 +740,7 @@ def _test_gale_transform(self, tester=None, **options):
 
             try:
                 import sage.graphs.graph
+                assert sage.graphs.graph  # to muffle pyflakes
             except ImportError:
                 pass
             else:
@@ -1008,6 +1008,7 @@ def _affine_hull_projection(self, *,
             except TypeError:
                 if not extend:
                     raise ValueError('the base ring needs to be extended; try with "extend=True"')
+                from sage.rings.qqbar import AA
                 M = matrix(AA, M)
                 A = M.gram_schmidt(orthonormal=orthonormal)[0]
                 if minimal:
@@ -1217,9 +1218,9 @@ def affine_hull_projection(self,
              A vertex at (2, 0, 0),
              A vertex at (1, 3/2, 0),
              A vertex at (1, 1/2, 4/3))
-            sage: A = S.affine_hull_projection(orthonormal=True, extend=True); A
+            sage: A = S.affine_hull_projection(orthonormal=True, extend=True); A                  # optional - sage.rings.number_field
             A 3-dimensional polyhedron in AA^3 defined as the convex hull of 4 vertices
-            sage: A.vertices()
+            sage: A.vertices()                                                                    # optional - sage.rings.number_field
             (A vertex at (0.7071067811865475?, 0.4082482904638630?, 1.154700538379252?),
              A vertex at (0.7071067811865475?, 1.224744871391589?, 0.?e-18),
              A vertex at (1.414213562373095?, 0.?e-18, 0.?e-18),
@@ -1228,11 +1229,11 @@ def affine_hull_projection(self,
         With the parameter ``minimal`` one can get a minimal base ring::
 
             sage: s = polytopes.simplex(3)
-            sage: s_AA = s.affine_hull_projection(orthonormal=True, extend=True)
-            sage: s_AA.base_ring()
+            sage: s_AA = s.affine_hull_projection(orthonormal=True, extend=True)                  # optional - sage.rings.number_field
+            sage: s_AA.base_ring()                                                                # optional - sage.rings.number_field
             Algebraic Real Field
-            sage: s_full = s.affine_hull_projection(orthonormal=True, extend=True, minimal=True)
-            sage: s_full.base_ring()
+            sage: s_full = s.affine_hull_projection(orthonormal=True, extend=True, minimal=True)  # optional - sage.rings.number_field
+            sage: s_full.base_ring()                                                              # optional - sage.rings.number_field
             Number Field in a with defining polynomial y^4 - 4*y^2 + 1 with a = 0.5176380902050415?
 
         More examples with the ``orthonormal`` parameter::
@@ -1486,21 +1487,25 @@ def _test_affine_hull_projection(self, tester=None, verbose=False, **options):
             # Avoid very long doctests.
             return
 
-        data_sets = [None]*4
-        data_sets[0] = self.affine_hull_projection(return_all_data=True)
+        try:
+            from sage.rings.qqbar import AA
+        except ImportError:
+            AA = None
+
+        data_sets = []
+        data_sets.append(self.affine_hull_projection(return_all_data=True))
         if self.is_compact():
-            data_sets[1] = self.affine_hull_projection(return_all_data=True,
-                                                       orthogonal=True,
-                                                       extend=True)
-            data_sets[2] = self.affine_hull_projection(return_all_data=True,
-                                                       orthonormal=True,
-                                                       extend=True)
-            data_sets[3] = self.affine_hull_projection(return_all_data=True,
-                                                       orthonormal=True,
-                                                       extend=True,
-                                                       minimal=True)
-        else:
-            data_sets = data_sets[:1]
+            data_sets.append(self.affine_hull_projection(return_all_data=True,
+                                                         orthogonal=True,
+                                                         extend=True))
+            if AA is not None:
+                data_sets.append(self.affine_hull_projection(return_all_data=True,
+                                                             orthonormal=True,
+                                                             extend=True))
+                data_sets.append(self.affine_hull_projection(return_all_data=True,
+                                                             orthonormal=True,
+                                                             extend=True,
+                                                             minimal=True))
 
         for i, data in enumerate(data_sets):
             if verbose:

src/sage/geometry/polyhedron/base7.py

@@ -35,7 +35,6 @@
 from sage.modules.free_module_element import vector
 from sage.rings.integer_ring import ZZ
 from sage.rings.rational_field import QQ
-from sage.rings.qqbar import AA
 from .base6 import Polyhedron_base6
 
 class Polyhedron_base7(Polyhedron_base6):
@@ -712,6 +711,7 @@ def volume(self, measure='ambient', engine='auto', **kwds):
             if Adet.is_square():
                 sqrt_Adet = Adet.sqrt()
             else:
+                from sage.rings.qqbar import AA
                 sqrt_Adet = AA(Adet).sqrt()
                 scaled_volume = AA(scaled_volume)
             return scaled_volume / sqrt_Adet
@@ -908,6 +908,7 @@ def integrate(self, function, measure='ambient', **kwds):
                 A = affine_hull_data.projection_linear_map.matrix()
                 Adet = (A.transpose() * A).det()
                 try:
+                    from sage.rings.qqbar import AA
                     Adet = AA.coerce(Adet)
                 except TypeError:
                     pass

src/sage/geometry/polyhedron/base_ZZ.py

@@ -17,7 +17,7 @@
 from sage.misc.cachefunc import cached_method
 from sage.modules.free_module_element import vector
 from .base_QQ import Polyhedron_QQ
-from sage.arith.all import gcd
+from sage.arith.misc import gcd
 
 
 #########################################################################

src/sage/geometry/polyhedron/parent.py

@@ -65,7 +65,7 @@ def Polyhedra(ambient_space_or_base_ring=None, ambient_dim=None, backend=None, *
     EXAMPLES::
 
         sage: from sage.geometry.polyhedron.parent import Polyhedra
-        sage: Polyhedra(AA, 3)
+        sage: Polyhedra(AA, 3)                                                  # optional - sage.rings.number_field
         Polyhedra in AA^3
         sage: Polyhedra(ZZ, 3)
         Polyhedra in ZZ^3
@@ -105,11 +105,11 @@ def Polyhedra(ambient_space_or_base_ring=None, ambient_dim=None, backend=None, *
         Traceback (most recent call last):
         ...
         ValueError: no default backend for computations with Real Field with 53 bits of precision
-        sage: Polyhedra(QQ[I], 2)
+        sage: Polyhedra(QQ[I], 2)                                               # optional - sage.rings.number_field
         Traceback (most recent call last):
         ...
         ValueError: invalid base ring: Number Field in I with defining polynomial x^2 + 1 with I = 1*I cannot be coerced to a real field
-        sage: Polyhedra(AA, 3, backend='polymake')  # optional - jupymake
+        sage: Polyhedra(AA, 3, backend='polymake')  # optional - jupymake       # optional - sage.rings.number_field
         Traceback (most recent call last):
         ...
         ValueError: the 'polymake' backend for polyhedron cannot be used with Algebraic Real Field
@@ -174,8 +174,9 @@ def Polyhedra(ambient_space_or_base_ring=None, ambient_dim=None, backend=None, *
     elif backend == 'polymake':
         base_field = base_ring.fraction_field()
         try:
-            from sage.interfaces.polymake import polymake
+            from sage.interfaces.polymake import polymake, PolymakeElement
             polymake_base_field = polymake(base_field)
+            assert isinstance(polymake_base_field, PolymakeElement)  # to muffle pyflakes
         except TypeError:
             raise ValueError(f"the 'polymake' backend for polyhedron cannot be used with {base_field}")
         return Polyhedra_polymake(base_field, ambient_dim, backend)
@@ -264,13 +265,13 @@ def list(self):
             sage: P.cardinality()
             +Infinity
 
-            sage: P = Polyhedra(AA, 0)
-            sage: P.category()
+            sage: P = Polyhedra(AA, 0)                                          # optional - sage.rings.number_field
+            sage: P.category()                                                  # optional - sage.rings.number_field
             Category of finite enumerated polyhedral sets over Algebraic Real Field
-            sage: P.list()
+            sage: P.list()                                                      # optional - sage.rings.number_field
             [The empty polyhedron in AA^0,
              A 0-dimensional polyhedron in AA^0 defined as the convex hull of 1 vertex]
-            sage: P.cardinality()
+            sage: P.cardinality()                                               # optional - sage.rings.number_field
             2
         """
         if self.ambient_dim():
@@ -495,6 +496,35 @@ def Hrepresentation_space(self):
             from sage.modules.free_module import FreeModule
             return FreeModule(self.base_ring(), self.ambient_dim()+1)
 
+    def _repr_base_ring(self):
+        """
+        Return an abbreviated string representation of the base ring.
+
+        EXAMPLES::
+
+            sage: from sage.geometry.polyhedron.parent import Polyhedra
+            sage: Polyhedra(QQ, 3)._repr_base_ring()
+            'QQ'
+            sage: K.<sqrt3> = NumberField(x^2 - 3, embedding=AA(3).sqrt())      # optional - sage.rings.number_field
+            sage: Polyhedra(K, 4)._repr_base_ring()                             # optional - sage.rings.number_field
+            '(Number Field in sqrt3 with defining polynomial x^2 - 3 with sqrt3 = 1.732050807568878?)'
+        """
+
+        if self.base_ring() is ZZ:
+            return 'ZZ'
+        if self.base_ring() is QQ:
+            return 'QQ'
+        if self.base_ring() is RDF:
+            return 'RDF'
+        try:
+            from sage.rings.qqbar import AA
+        except ImportError:
+            pass
+        else:
+            if self.base_ring() is AA:
+                return 'AA'
+        return '({0})'.format(self.base_ring())
+
     def _repr_ambient_module(self):
         """
         Return an abbreviated string representation of the ambient
@@ -509,21 +539,11 @@ def _repr_ambient_module(self):
             sage: from sage.geometry.polyhedron.parent import Polyhedra
             sage: Polyhedra(QQ, 3)._repr_ambient_module()
             'QQ^3'
-            sage: K.<sqrt3> = NumberField(x^2 - 3, embedding=AA(3).sqrt())
-            sage: Polyhedra(K, 4)._repr_ambient_module()
+            sage: K.<sqrt3> = NumberField(x^2 - 3, embedding=AA(3).sqrt())      # optional - sage.rings.number_field
+            sage: Polyhedra(K, 4)._repr_ambient_module()                        # optional - sage.rings.number_field
             '(Number Field in sqrt3 with defining polynomial x^2 - 3 with sqrt3 = 1.732050807568878?)^4'
         """
-        from sage.rings.qqbar import AA
-        if self.base_ring() is ZZ:
-            s = 'ZZ'
-        elif self.base_ring() is QQ:
-            s = 'QQ'
-        elif self.base_ring() is RDF:
-            s = 'RDF'
-        elif self.base_ring() is AA:
-            s = 'AA'
-        else:
-            s = '({0})'.format(self.base_ring())
+        s = self._repr_base_ring()
         s += '^' + repr(self.ambient_dim())
         return s
 
@@ -602,11 +622,11 @@ def _element_constructor_(self, *args, **kwds):
 
         When the parent of the object is not ``self``, the default is not to copy::
 
-            sage: Q = P.base_extend(AA)
-            sage: q = Q._element_constructor_(p)
-            sage: q is p
+            sage: Q = P.base_extend(AA)                                         # optional - sage.rings.number_field
+            sage: q = Q._element_constructor_(p)                                # optional - sage.rings.number_field
+            sage: q is p                                                        # optional - sage.rings.number_field
             False
-            sage: q = Q._element_constructor_(p, copy=False)
+            sage: q = Q._element_constructor_(p, copy=False)                    # optional - sage.rings.number_field
             Traceback (most recent call last):
             ...
             ValueError: you need to make a copy when changing the parent
@@ -693,9 +713,10 @@ def _element_constructor_polyhedron(self, polyhedron, **kwds):
             sage: P(p)
             A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 4 vertices
 
-            sage: P = Polyhedra(AA, 3, backend='field')
-            sage: p = Polyhedron(vertices=[(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1)])
-            sage: P(p)
+            sage: P = Polyhedra(AA, 3, backend='field')                         # optional - sage.rings.number_field
+            sage: vertices = [(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1)]
+            sage: p = Polyhedron(vertices=vertices)                             # optional - sage.rings.number_field
+            sage: P(p)                                                          # optional - sage.rings.number_field
             A 3-dimensional polyhedron in AA^3 defined as the convex hull of 4 vertices
         """
         Vrep = None
@@ -1234,9 +1255,9 @@ def does_backend_handle_base_ring(base_ring, backend):
         sage: from sage.geometry.polyhedron.parent import does_backend_handle_base_ring
         sage: does_backend_handle_base_ring(QQ, 'ppl')
         True
-        sage: does_backend_handle_base_ring(QQ[sqrt(5)], 'ppl')
+        sage: does_backend_handle_base_ring(QQ[sqrt(5)], 'ppl')                 # optional - sage.rings.number_field
         False
-        sage: does_backend_handle_base_ring(QQ[sqrt(5)], 'field')
+        sage: does_backend_handle_base_ring(QQ[sqrt(5)], 'field')               # optional - sage.rings.number_field
         True
     """
     try: