Fixing the doc and != for polynomial injection maps.

src/sage/rings/polynomial/ore_polynomial_element.pyx

@@ -20,15 +20,15 @@ AUTHORS:
 - Xavier Caruso (2020-05)
 """
 
-#############################################################################
+# ***************************************************************************
 #    Copyright (C) 2020 Xavier Caruso <xavier.caruso@normalesup.org>
 #
 #    This program is free software: you can redistribute it and/or modify
 #    it under the terms of the GNU General Public License as published by
 #    the Free Software Foundation, either version 2 of the License, or
 #    (at your option) any later version.
 #                  http://www.gnu.org/licenses/
-#****************************************************************************
+# ***************************************************************************
 
 import re
 from cysignals.signals cimport sig_check

src/sage/rings/polynomial/ore_polynomial_ring.py

@@ -1,9 +1,10 @@
 r"""
 Univariate Ore Polynomial Rings
 
-This module provides the :class:`~sage.rings.polynomial.ore_polynomial_ring.OrePolynomialRing`.
-which constructs a general dense univariate Ore polynomial ring over a commutative base with
-equipped with an endomorphism and/or a derivation.
+This module provides the
+:class:`~sage.rings.polynomial.ore_polynomial_ring.OrePolynomialRing`,
+which constructs a general dense univariate Ore polynomial ring over a
+commutative base with equipped with an endomorphism and/or a derivation.
 
 AUTHOR:
 
@@ -68,14 +69,14 @@ class OrePolynomialRing(UniqueRepresentation, Algebra):
 
     - ``names`` -- a string or a list of strings
 
-    - ``sparse`` -- a boolean (default: ``False``). Currently not supported.
+    - ``sparse`` -- a boolean (default: ``False``); currently not supported
 
     EXAMPLES:
 
     .. RUBRIC:: The case of a twisting endomorphism
 
     We create the Ore ring `\GF{5^3}[x, \text{Frob}]` where Frob is the
-    Frobenius endomorphism.
+    Frobenius endomorphism::
 
         sage: k.<a> = GF(5^3)
         sage: Frob = k.frobenius_endomorphism()
@@ -96,7 +97,7 @@ class OrePolynomialRing(UniqueRepresentation, Algebra):
 
     We emphasize that it is necessary to repeat the name of the variable
     in the right hand side. Indeed, the following fails (it is interpreted
-    by Sage as a classical polynomial ring with variable name ``Frob``).
+    by Sage as a classical polynomial ring with variable name ``Frob``)::
 
         sage: T.<x> = k[Frob]
         Traceback (most recent call last):
@@ -147,13 +148,15 @@ class OrePolynomialRing(UniqueRepresentation, Algebra):
 
         sage: F.<u> = Qq(3^2)
         sage: sigma = F.frobenius_endomorphism(); sigma
-        Frobenius endomorphism on 3-adic Unramified Extension Field in u defined by x^2 + 2*x + 2 lifting u |--> u^3 on the residue field
+        Frobenius endomorphism on 3-adic Unramified Extension Field in u
+         defined by x^2 + 2*x + 2 lifting u |--> u^3 on the residue field
         sage: der = F.derivation(3, twist=sigma); der
         (3 + O(3^21))*([Frob] - id)
 
         sage: M.<X> = F['X', der]
         sage: M
-        Ore Polynomial Ring in X over 3-adic Unramified Extension Field in u defined by x^2 + 2*x + 2 twisted by Frob and (3 + O(3^21))*([Frob] - id)
+        Ore Polynomial Ring in X over 3-adic Unramified Extension Field in u
+         defined by x^2 + 2*x + 2 twisted by Frob and (3 + O(3^21))*([Frob] - id)
 
     We emphasize that we only need to pass in the twisted derivation as
     it already contains in it the datum of the twisting endomorphism.
@@ -166,6 +169,8 @@ class OrePolynomialRing(UniqueRepresentation, Algebra):
 
     .. RUBRIC:: Examples of variable name context
 
+    Consider the following::
+
         sage: R.<t> = ZZ[]
         sage: sigma = R.hom([t+1])
         sage: S.<x> = SkewPolynomialRing(R, sigma); S
@@ -187,7 +192,7 @@ class OrePolynomialRing(UniqueRepresentation, Algebra):
     .. RUBRIC:: Uniqueness and immutability
 
     In Sage, there is exactly one Ore polynomial ring for each quadruple
-    (base ring, twisting morphism, twisting derivation, name of the variable).
+    (base ring, twisting morphism, twisting derivation, name of the variable)::
 
         sage: k.<a> = GF(7^3)
         sage: Frob = k.frobenius_endomorphism()
@@ -642,12 +647,12 @@ def characteristic(self):
         """
         return self.base_ring().characteristic()
 
-
     @cached_method
     def twisting_morphism(self, n=1):
         r"""
-        Return the twisting endomorphism defining this Ore polynomial ring iterated ``n`` times
-        or ``None`` if this Ore polynomial ring is not twisted by an endomorphism.
+        Return the twisting endomorphism defining this Ore polynomial ring
+        iterated ``n`` times or ``None`` if this Ore polynomial ring is not
+        twisted by an endomorphism.
 
         INPUT:
 
@@ -676,7 +681,8 @@ def twisting_morphism(self, n=1):
             sage: T.twisting_morphism(-1)
             Frobenius endomorphism t |--> t^(5^2) on Finite Field in t of size 5^3
 
-        Sometimes it fails, even if the twisting morphism is actually invertible::
+        Sometimes it fails, even if the twisting morphism is
+        actually invertible::
 
             sage: S.twisting_morphism(-1)
             Traceback (most recent call last):
@@ -720,10 +726,11 @@ def twisting_morphism(self, n=1):
 
     def twist_map(self, n=1):
         r"""
-        Return the twisting endomorphism defining this Ore polynomial ring iterated ``n`` times
-        or ``None`` if this Ore polynomial ring is not twisted by an endomorphism.
+        Return the twisting endomorphism defining this Ore polynomial ring
+        iterated ``n`` times or ``None`` if this Ore polynomial ring is not
+        twisted by an endomorphism.
 
-        This method is deprecated. You should use :meth:`twisting_morphism` instead.
+        This method is deprecated. Use :meth:`twisting_morphism` instead.
 
         INPUT:
 
@@ -770,7 +777,6 @@ def twisting_derivation(self):
         """
         return self._derivation
 
-
     @cached_method
     def gen(self, n=0):
         r"""
@@ -867,8 +873,8 @@ def is_exact(self):
 
     def is_sparse(self):
         r"""
-        Return ``True`` if the elements of this Ore polynomial ring are sparsely
-        represented.
+        Return ``True`` if the elements of this Ore polynomial ring are
+        sparsely represented.
 
         .. WARNING::
 
@@ -931,15 +937,17 @@ def random_element(self, degree=2, monic=False, *args, **kwds):
             sage: S.random_element(monic=True)  # random
             x^2 + (2*t^2 + t + 1)*x + 3*t^2 + 3*t + 2
 
-        Use ``degree`` to obtain polynomials of higher degree
+        Use ``degree`` to obtain polynomials of higher degree::
 
             sage: p = S.random_element(degree=5)   # random
             (t^2 + 3*t)*x^4 + (4*t + 4)*x^3 + (4*t^2 + 4*t)*x^2 + (2*t^2 + 1)*x + 3
+            sage: p.degree() <= 5
+            True
 
         When ``monic`` is ``False``, the returned Ore polynomial may have
         a degree less than ``degree`` (it happens when the random leading
-        coefficient is zero). However, if ``monic`` is ``True``, this can't
-        happen::
+        coefficient is zero). However, if ``monic`` is ``True``, this
+        cannot happen::
 
             sage: p = S.random_element(degree=4, monic=True)
             sage: p.leading_coefficient() == S.base_ring().one()
@@ -1049,3 +1057,4 @@ def is_commutative(self):
             True
         """
         return self._morphism is None and self._derivation is None
+

src/sage/rings/polynomial/skew_polynomial_element.pyx

@@ -36,10 +36,9 @@ AUTHORS:
 
 - Johan Rosenkilde (2016-08-03): changes for bug fixes, docstring and
   doctest errors
-
 """
 
-#############################################################################
+# ***************************************************************************
 #    Copyright (C) 2012 Xavier Caruso <xavier.caruso@normalesup.org>
 #
 #    This program is free software: you can redistribute it and/or modify
@@ -345,7 +344,8 @@ cdef class SkewPolynomial_generic_dense(OrePolynomial_generic_dense):
             sage: a.operator_eval(t)
             2*t^2 + 2*t + 3
 
-        Evaluation points outside the base ring is usually not possible due to the twisting morphism::
+        Evaluation points outside the base ring is usually not possible
+        due to the twisting morphism::
 
             sage: R.<t> = QQ[]
             sage: sigma = R.hom([t+1])
@@ -354,7 +354,9 @@ cdef class SkewPolynomial_generic_dense(OrePolynomial_generic_dense):
             sage: a.operator_eval(1/t)
             Traceback (most recent call last):
             ...
-            TypeError: 1/t fails to convert into the map's domain Univariate Polynomial Ring in t over Rational Field, but a `pushforward` method is not properly implemented
+            TypeError: 1/t fails to convert into the map's domain
+             Univariate Polynomial Ring in t over Rational Field,
+             but a `pushforward` method is not properly implemented
         """
         cdef RingHomomorphism sigma = self._parent.twisting_morphism()
         cdef list coefficients = self.list()
@@ -414,7 +416,7 @@ cdef class SkewPolynomial_generic_dense(OrePolynomial_generic_dense):
         return r
 
     def multi_point_evaluation(self, eval_pts):
-        """
+        r"""
         Evaluate ``self`` at list of evaluation points.
 
         INPUT:
@@ -431,7 +433,7 @@ cdef class SkewPolynomial_generic_dense(OrePolynomial_generic_dense):
             repeatedly. If fast skew polynomial multiplication is available, an
             asymptotically faster method is possible using standard divide and
             conquer techniques and
-            :meth:`sage.rings.polynomial.skew_polynomial_ring.SkewPolynomialRing_general.minimal_vanishing_polynomial`.
+            :meth:`~sage.rings.polynomial.skew_polynomial_ring.SkewPolynomialRing.minimal_vanishing_polynomial`.
 
         EXAMPLES::
 
@@ -671,3 +673,4 @@ cdef class SkewPolynomial_generic_dense(OrePolynomial_generic_dense):
             q.append(c)
         q.reverse()
         return (self._new_c(q, parent), self._new_c(a[:db], parent, 1))
+