some details in multi_polynomial base

src/sage/rings/polynomial/multi_polynomial_ring_base.pyx

@@ -41,10 +41,11 @@ cdef class MPolynomialRing_base(sage.rings.ring.CommutativeRing):
 
             sage: R.<x,y> = ZZ['x,y']; R
             Multivariate Polynomial Ring in x, y over Integer Ring
-            sage: class CR(CommutativeRing):
+            sage: cat = Rings().Commutative()
+            sage: class CR(Parent):
             ....:     def __init__(self):
-            ....:         CommutativeRing.__init__(self,self)
-            ....:     def __call__(self,x):
+            ....:         Parent.__init__(self, self, category=cat)
+            ....:     def __call__(self, x):
             ....:         return None
             sage: cr = CR()
             sage: cr.is_commutative()
@@ -668,7 +669,7 @@ cdef class MPolynomialRing_base(sage.rings.ring.CommutativeRing):
 
     def repr_long(self):
         """
-        Return structured string representation of self.
+        Return structured string representation of ``self``.
 
         EXAMPLES::
 
@@ -758,7 +759,6 @@ cdef class MPolynomialRing_base(sage.rings.ring.CommutativeRing):
             Order: Graded Reverse Lexicographical
             Variables: T, W
 
-
             sage: # optional - magma
             sage: magma(PolynomialRing(GF(7),4, 'x'))
             Polynomial ring of rank 4 over GF(7)
@@ -868,12 +868,12 @@ cdef class MPolynomialRing_base(sage.rings.ring.CommutativeRing):
 
     def gen(self, n=0):
         if n < 0 or n >= self._ngens:
-            raise ValueError("Generator not defined.")
+            raise ValueError("generator not defined")
         return self._gens[int(n)]
 
     def variable_names_recursive(self, depth=sage.rings.infinity.infinity):
         r"""
-        Returns the list of variable names of this and its base rings, as if
+        Return the list of variable names of this and its base rings, as if
         it were a single multi-variate polynomial.
 
         EXAMPLES::
@@ -883,7 +883,6 @@ cdef class MPolynomialRing_base(sage.rings.ring.CommutativeRing):
             ('x', 'y', 'z', 'w')
             sage: R.variable_names_recursive(3)
             ('y', 'z', 'w')
-
         """
         if depth <= 0:
             all = ()
@@ -901,7 +900,8 @@ cdef class MPolynomialRing_base(sage.rings.ring.CommutativeRing):
 
     def _mpoly_base_ring(self, vars=None):
         """
-        Returns the base ring if this is viewed as a polynomial ring over vars.
+        Return the base ring if this is viewed as a polynomial ring over vars.
+
         See also MPolynomial._mpoly_dict_recursive.
         """
         if vars is None:
@@ -1015,10 +1015,7 @@ cdef class MPolynomialRing_base(sage.rings.ring.CommutativeRing):
         if not comb:
             return (d,)
         monomial = [comb[0]]
-        res = []
-        for j in range(n - 2):
-            res.append(comb[j + 1] - comb[j] - 1)
-        monomial += res
+        monomial.extend(comb[j + 1] - comb[j] - 1 for j in range(n - 2))
         monomial.append(n + d - 1 - comb[-1] - 1)
         return tuple(monomial)
 
@@ -1234,7 +1231,6 @@ cdef class MPolynomialRing_base(sage.rings.ring.CommutativeRing):
             True
             sage: R.random_element().parent() == R
             True
-
         """
         k = self.base_ring()
         n = self.ngens()
@@ -1243,7 +1239,7 @@ cdef class MPolynomialRing_base(sage.rings.ring.CommutativeRing):
 
         # Note that 'terms' could be None while 'total' is a
         # nonnegative integer, so the comparison 'terms > total' could
-        # fail in Python 3.
+        # fail
         if terms and terms > total:
             terms = total
 
@@ -1526,13 +1522,14 @@ cdef class MPolynomialRing_base(sage.rings.ring.CommutativeRing):
 
     def macaulay_resultant(self, *args, **kwds):
         r"""
-        This is an implementation of the Macaulay resultant. It computes
-        the resultant of universal polynomials as well as polynomials
-        with constant coefficients. This is a project done in
-        sage days 55. It's based on the implementation in Maple by
-        Manfred Minimair, which in turn is based on the references listed below:
-        It calculates the Macaulay resultant for a list of polynomials,
-        up to sign!
+        Return the Macaulay resultant.
+
+        This computes the resultant of universal polynomials as well as
+        polynomials with constant coefficients. This is a project done
+        in sage days 55. It is based on the implementation in Maple by
+        Manfred Minimair, which in turn is based on the references
+        listed below. It calculates the Macaulay resultant for a list
+        of polynomials, up to sign!
 
         REFERENCES:
 
@@ -1562,6 +1559,7 @@ cdef class MPolynomialRing_base(sage.rings.ring.CommutativeRing):
         - the Macaulay resultant, an element of the base ring of ``self``
 
         .. TODO::
+
             Working with sparse matrices should usually give faster results,
             but with the current implementation it actually works slower.
             There should be a way to improve performance with regards to this.
@@ -1684,7 +1682,7 @@ cdef class MPolynomialRing_base(sage.rings.ring.CommutativeRing):
         else:
             flist = args
 
-        if len(flist) <= 0:
+        if len(flist) == 0:
             raise TypeError('input list should contain at least 1 polynomial')
         if not all(f.is_homogeneous() for f in flist):
             raise TypeError('resultant for non-homogeneous polynomials is not supported')