gh-37869: `sage.geometry.polyhedron`: Remove deprecated code
    
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Remove code deprecated in:
- #32592 (2021)
- #33646 (2022)
- #31834 (2021)

### 📝 Checklist

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- [x] The title is concise and informative.
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- [x] I have linked a relevant issue or discussion.
- [ ] I have created tests covering the changes.
- [ ] I have updated the documentation and checked the documentation
preview.

### ⌛ Dependencies

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URL: #37869
Reported by: Matthias Köppe
Reviewer(s): Kwankyu Lee

Author: Release Manager

src/sage/geometry/polyhedron/backend_cdd.py

@@ -22,9 +22,6 @@
 from .base import Polyhedron_base
 from .base_QQ import Polyhedron_QQ
 
-from sage.misc.lazy_import import lazy_import
-lazy_import('sage.geometry.polyhedron.backend_cdd_rdf', 'Polyhedron_RDF_cdd', deprecation=32592)
-
 
 class Polyhedron_cdd(Polyhedron_base):
     r"""

src/sage/geometry/polyhedron/base3.py

@@ -372,7 +372,7 @@ def _test_combinatorial_polyhedron(self, tester=None, **options):
                                                        prefix=tester._prefix+"  ")
         tester.info(tester._prefix + " ", newline=False)
 
-    def face_generator(self, face_dimension=None, algorithm=None, **kwds):
+    def face_generator(self, face_dimension=None, algorithm=None):
         r"""
         Return an iterator over the faces of given dimension.
 
@@ -590,22 +590,6 @@ def face_generator(self, face_dimension=None, algorithm=None, **kwds):
             sage: f.ambient_Hrepresentation()
             (An equation (1, 1, 1) x - 6 == 0,)
 
-        The ``dual`` keyword is deprecated::
-
-             sage: P = polytopes.hypercube(4)
-             sage: list(P.face_generator(dual=False))[:4]
-             doctest:...: DeprecationWarning: the keyword dual is deprecated; use algorithm instead
-             See https://github.com/sagemath/sage/issues/33646 for details.
-             [A 4-dimensional face of a Polyhedron in ZZ^4 defined as the convex hull of 16 vertices,
-              A -1-dimensional face of a Polyhedron in ZZ^4,
-              A 3-dimensional face of a Polyhedron in ZZ^4 defined as the convex hull of 8 vertices,
-              A 3-dimensional face of a Polyhedron in ZZ^4 defined as the convex hull of 8 vertices]
-             sage: list(P.face_generator(True))[:4]
-             [A 1-dimensional face of a Polyhedron in ZZ^4 defined as the convex hull of 2 vertices,
-              A 1-dimensional face of a Polyhedron in ZZ^4 defined as the convex hull of 2 vertices,
-              A 1-dimensional face of a Polyhedron in ZZ^4 defined as the convex hull of 2 vertices,
-              A 1-dimensional face of a Polyhedron in ZZ^4 defined as the convex hull of 2 vertices]
-
         Check that we catch incorrect algorithms::
 
              sage: list(P.face_generator(2, algorithm='integrate'))[:4]
@@ -618,19 +602,9 @@ def face_generator(self, face_dimension=None, algorithm=None, **kwds):
             dual = False
         elif algorithm == 'dual':
             dual = True
-        elif algorithm in (False, True):
-            from sage.misc.superseded import deprecation
-            deprecation(33646, "the keyword dual is deprecated; use algorithm instead")
-            dual = algorithm
         elif algorithm is not None:
             raise ValueError("algorithm must be 'primal', 'dual' or None")
 
-        if kwds:
-            from sage.misc.superseded import deprecation
-            deprecation(33646, "the keyword dual is deprecated; use algorithm instead")
-            if 'dual' in kwds and dual is None:
-                dual = kwds['dual']
-
         from sage.geometry.polyhedron.combinatorial_polyhedron.face_iterator import FaceIterator_geom
         return FaceIterator_geom(self, output_dimension=face_dimension, dual=dual)
 

src/sage/geometry/polyhedron/combinatorial_polyhedron/base.pxd

@@ -34,7 +34,6 @@ cdef class CombinatorialPolyhedron(SageObject):
     cdef tuple Vrep(self)
     cdef tuple facet_names(self)
     cdef tuple equations(self)
-    cdef tuple equalities(self)
     cdef unsigned int n_Vrepresentation(self) noexcept
     cdef unsigned int n_Hrepresentation(self) noexcept
     cdef bint is_bounded(self) noexcept

src/sage/geometry/polyhedron/combinatorial_polyhedron/base.pyx

@@ -1359,7 +1359,7 @@ cdef class CombinatorialPolyhedron(SageObject):
         adjacency_matrix.set_immutable()
         return adjacency_matrix
 
-    def ridges(self, add_equations=False, names=True, add_equalities=False, algorithm=None):
+    def ridges(self, add_equations=False, names=True, algorithm=None):
         r"""
         Return the ridges.
 
@@ -1453,21 +1453,7 @@ cdef class CombinatorialPolyhedron(SageObject):
             sage: C = CombinatorialPolyhedron(polytopes.simplex())
             sage: C.ridges(names=False, add_equations=True)
             ((2, 3), (1, 3), (0, 3), (1, 2), (0, 2), (0, 1))
-
-        The keyword ``add_equalities`` is deprecated::
-
-            sage: C = CombinatorialPolyhedron(polytopes.simplex())
-            sage: r = C.ridges(add_equations=True)
-            sage: r1 = C.ridges(add_equalities=True)
-            doctest:...: DeprecationWarning: the keyword ``add_equalities`` is deprecated; use ``add_equations``
-            See https://github.com/sagemath/sage/issues/31834 for details.
-            sage: r == r1
-            True
         """
-        if add_equalities:
-            from sage.misc.superseded import deprecation
-            deprecation(31834, "the keyword ``add_equalities`` is deprecated; use ``add_equations``", 3)
-            add_equations = True
         self._compute_ridges(self._algorithm_to_dual(algorithm))
         cdef size_t n_ridges = self._ridges.length
 
@@ -2673,7 +2659,7 @@ cdef class CombinatorialPolyhedron(SageObject):
         """
         return self.face_generator().meet_of_Hrep(*indices)
 
-    def face_generator(self, dimension=None, algorithm=None, **kwds):
+    def face_generator(self, dimension=None, algorithm=None):
         r"""
         Iterator over all proper faces of specified dimension.
 
@@ -2760,35 +2746,14 @@ cdef class CombinatorialPolyhedron(SageObject):
             (A ray in the direction (1, 0), A vertex at (1, 0))
             (A ray in the direction (0, 1), A vertex at (0, 1))
 
-        TESTS:
-
-        The kewword ``dual`` is deprecated::
-
-            sage: C = CombinatorialPolyhedron([[0,1,2],[0,1,3],[0,2,3],[1,2,3]])
-            sage: it = C.face_generator(1, False)
-            doctest:...: DeprecationWarning: the keyword dual is deprecated; use algorithm instead
-            See https://github.com/sagemath/sage/issues/33646 for details.
-            sage: it = C.face_generator(1, dual=True)
-
         .. SEEALSO::
 
             :class:`~sage.geometry.polyhedron.combinatorial_polyhedron.face_iterator.FaceIterator`,
             :class:`~sage.geometry.polyhedron.combinatorial_polyhedron.combinatorial_face.CombinatorialFace`.
         """
         cdef int dual
 
-        if algorithm in (False, True):
-            from sage.misc.superseded import deprecation
-            deprecation(33646, "the keyword dual is deprecated; use algorithm instead")
-            dual = int(algorithm)
-        else:
-            dual = self._algorithm_to_dual(algorithm)
-
-        if kwds:
-            from sage.misc.superseded import deprecation
-            deprecation(33646, "the keyword dual is deprecated; use algorithm instead")
-            if 'dual' in kwds and dual == -1 and kwds['dual'] in (False, True):
-                dual = int(kwds['dual'])
+        dual = self._algorithm_to_dual(algorithm)
 
         if dual == -1:
             # Determine the faster way, to iterate through all faces.
@@ -3273,11 +3238,6 @@ cdef class CombinatorialPolyhedron(SageObject):
         """
         return self._equations
 
-    cdef tuple equalities(self):
-        from sage.misc.superseded import deprecation
-        deprecation(31834, "the method equalities of CombinatorialPolyhedron is deprecated; use equations", 3)
-        return self.equations()
-
     cdef unsigned int n_Vrepresentation(self) noexcept:
         r"""
         Return the number of elements in the Vrepresentation.