Trac #23337: Use variable names instead of symbolic variables

Where possible, use {{{QQ["x"]}}} instead of {{{QQ[x]}}} to construct
polynomial rings. After #10483, the latter will be deprecated.

URL: https://trac.sagemath.org/23337
Reported by: jdemeyer
Ticket author(s): Simon King, Ralf Stephan, Jeroen Demeyer
Reviewer(s): Jeroen Demeyer, Travis Scrimshaw

Author: Release Manager

src/sage/combinat/ncsf_qsym/qsym.py

@@ -691,7 +691,7 @@ def from_polynomial(self, f, check=True):
             if I not in z:
                 z[I] = c
         out = self.Monomial()._from_dict(z)
-        if check and out.expand(f.parent().ngens(), f.parent().gens()) != f:
+        if check and out.expand(f.parent().ngens(), f.parent().variable_names()) != f:
             raise ValueError("%s is not a quasi-symmetric polynomial" % f)
         return out
 

src/sage/combinat/root_system/root_lattice_realization_algebras.py

@@ -256,13 +256,13 @@ def demazure_operators(self):
             This is indeed a Schur function::
 
                 sage: s = SymmetricFunctions(QQ).s()
-                sage: s[2,1].expand(3, P.gens())
+                sage: s[2,1].expand(3, P.variable_names())
                 x^2*y + x*y^2 + x^2*z + 2*x*y*z + y^2*z + x*z^2 + y*z^2
 
             Let us check this systematically on Schur functions of degree 6::
 
                 sage: for p in Partitions(6, max_length=3).list():
-                ....:     assert s.monomial(p).expand(3, P.gens()) == pi0(KL.monomial(L(tuple(p)))).expand(P.gens())
+                ....:     assert s.monomial(p).expand(3, P.variable_names()) == pi0(KL.monomial(L(tuple(p)))).expand(P.gens())
 
             We check systematically that these operators satisfy the Iwahori-Hecke algebra relations::
 

src/sage/combinat/sf/monomial.py

@@ -174,7 +174,7 @@ def from_polynomial(self, f, check=True):
         out = self.sum_of_terms((Partition(e), c)
                                 for (e,c) in six.iteritems(f.dict())
                                 if tuple(sorted(e)) == tuple(reversed(e)))
-        if check and out.expand(f.parent().ngens(),f.parent().gens()) != f:
+        if check and out.expand(f.parent().ngens(),f.parent().variable_names()) != f:
             raise ValueError("%s is not a symmetric polynomial"%f)
         return out
 

src/sage/combinat/subset.py

@@ -1144,7 +1144,7 @@ def generating_serie(self,variable='x'):
             sage: l = [1,1,1,1,2,2,3]
             sage: for k in range(len(l)):
             ....:    S = Subsets(l,k,submultiset=True)
-            ....:    print(S.generating_serie(x) == S.cardinality()*x**k)
+            ....:    print(S.generating_serie('x') == S.cardinality()*x**k)
             True
             True
             True

src/sage/functions/piecewise.py

@@ -873,7 +873,7 @@ def critical_points(cls, self, parameters, variable):
                 True
             """
             from sage.calculus.calculus import maxima
-            x = QQ[self.default_variable()].gen()
+            x = self.default_variable()
             crit_pts = []
             for domain, f in parameters:
                 for interval in domain:

src/sage/matrix/compute_J_ideal.py

@@ -195,7 +195,7 @@ def lifting(p, t, A, G):
 
 
     DX = A.parent().base()
-    (X,) = DX.gens()
+    (X,) = DX.variable_names()
     D = DX.base_ring()
     d = A.ncols()
     c = A.nrows()

src/sage/matrix/matrix2.pyx

@@ -1647,7 +1647,7 @@ cdef class Matrix(Matrix1):
         # resulted in further exceptions/ errors.
         from sage.symbolic.ring import is_SymbolicExpressionRing
 
-        var = R('A0123456789') if is_SymbolicExpressionRing(R) else 'x'
+        var = 'A0123456789' if is_SymbolicExpressionRing(R) else 'x'
         try:
             c = self.charpoly(var, algorithm="df")[0]
         except ValueError:

src/sage/matrix/matrix_symbolic_dense.pyx

@@ -386,6 +386,10 @@ cdef class Matrix_symbolic_dense(Matrix_generic_dense):
         """
         Compute the characteristic polynomial of self, using maxima.
 
+        INPUT:
+
+        - ``var`` - (default: 'x') name of variable of charpoly
+
         EXAMPLES::
 
             sage: M = matrix(SR, 2, 2, var('a,b,c,d'))

src/sage/misc/latex.py

@@ -1905,7 +1905,7 @@ def __call__(self, x, combine_all=False):
             sage: from sage.misc.latex import MathJax
             sage: MathJax()(3)
             <html><script type="math/tex; mode=display">\newcommand{\Bold}[1]{\mathbf{#1}}3</script></html>
-            sage: str(MathJax().eval(ZZ[x], mode='display')) == str(MathJax()(ZZ[x]))
+            sage: str(MathJax().eval(ZZ['x'], mode='display')) == str(MathJax()(ZZ['x']))
             True
         """
         return self.eval(x, combine_all=combine_all)

src/sage/modules/free_module_element.pyx

@@ -2803,28 +2803,28 @@ cdef class FreeModuleElement(Vector):   # abstract base class
 
             sage: parent(vector(QQ,[1,2,3,4]).pairwise_product(vector(ZZ['x'],[1,2,3,4])))
             Ambient free module of rank 4 over the principal ideal domain Univariate Polynomial Ring in x over Rational Field
-            sage: parent(vector(ZZ[x],[1,2,3,4]).pairwise_product(vector(QQ,[1,2,3,4])))
+            sage: parent(vector(ZZ['x'],[1,2,3,4]).pairwise_product(vector(QQ,[1,2,3,4])))
             Ambient free module of rank 4 over the principal ideal domain Univariate Polynomial Ring in x over Rational Field
 
         ::
 
             sage: parent(vector(QQ,[1,2,3,4]).pairwise_product(vector(ZZ['x']['y'],[1,2,3,4])))
             Ambient free module of rank 4 over the integral domain Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
-            sage: parent(vector(ZZ[x][y],[1,2,3,4]).pairwise_product(vector(QQ,[1,2,3,4])))
+            sage: parent(vector(ZZ['x']['y'],[1,2,3,4]).pairwise_product(vector(QQ,[1,2,3,4])))
             Ambient free module of rank 4 over the integral domain Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
 
         ::
 
             sage: parent(vector(QQ['x'],[1,2,3,4]).pairwise_product(vector(ZZ['x']['y'],[1,2,3,4])))
             Ambient free module of rank 4 over the integral domain Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
-            sage: parent(vector(ZZ[x][y],[1,2,3,4]).pairwise_product(vector(QQ['x'],[1,2,3,4])))
+            sage: parent(vector(ZZ['x']['y'],[1,2,3,4]).pairwise_product(vector(QQ['x'],[1,2,3,4])))
             Ambient free module of rank 4 over the integral domain Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
 
         ::
 
             sage: parent(vector(QQ['y'],[1,2,3,4]).pairwise_product(vector(ZZ['x']['y'],[1,2,3,4])))
             Ambient free module of rank 4 over the integral domain Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
-            sage: parent(vector(ZZ[x][y],[1,2,3,4]).pairwise_product(vector(QQ['y'],[1,2,3,4])))
+            sage: parent(vector(ZZ['x']['y'],[1,2,3,4]).pairwise_product(vector(QQ['y'],[1,2,3,4])))
             Ambient free module of rank 4 over the integral domain Univariate Polynomial Ring in y over Univariate Polynomial Ring in x over Rational Field
 
         ::

src/sage/numerical/interactive_simplex_method.py

@@ -4116,7 +4116,7 @@ def add_row(self, nonbasic_coefficients, constant, basic_variable=None):
             basic_variable = str(basic_variable)
 
         R = PolynomialRing(
-            BR, list(B.base_ring().gens()) + [basic_variable], order="neglex")
+            BR, list(B.base_ring().variable_names()) + [basic_variable], order="neglex")
         B = list(B) + [basic_variable]
         B = map(R, B)
         N = map(R, N)

src/sage/rings/complex_arb.pyx

@@ -104,7 +104,7 @@ Automatic coercions work as expected::
 
 TESTS::
 
-    sage: polygen(CBF, x)^3
+    sage: polygen(CBF, 'x')^3
     x^3
 
 ::

src/sage/rings/ideal.py

@@ -963,9 +963,10 @@ def category(self):
 
         EXAMPLES::
 
+            sage: P.<x> = ZZ[]
             sage: I = ZZ.ideal(7)
-            sage: J = ZZ[x].ideal(7,x)
-            sage: K = ZZ[x].ideal(7)
+            sage: J = P.ideal(7,x)
+            sage: K = P.ideal(7)
             sage: I.category()
             Category of ring ideals in Integer Ring
             sage: J.category()
@@ -1140,7 +1141,7 @@ def __repr__(self):
 
         EXAMPLES::
 
-            sage: R = ZZ[x]
+            sage: R.<x> = ZZ[]
             sage: I = R.ideal(x)
             sage: I # indirect doctest
             Principal ideal (x) of Univariate Polynomial Ring in x over Integer Ring
@@ -1158,7 +1159,7 @@ def is_principal(self):
         Note that Sage automatically coerces ideals into
         principal ideals during initialization::
 
-            sage: R = ZZ[x]
+            sage: R.<x> = ZZ[]
             sage: I = R.ideal(x)
             sage: J = R.ideal(2,x)
             sage: K = R.base_extend(QQ).ideal(2,x)
@@ -1196,7 +1197,7 @@ def gen(self):
         Note that the generator belongs to the ring from which the ideal
         was initialized::
 
-            sage: R = ZZ[x]
+            sage: R.<x> = ZZ[]
             sage: I = R.ideal(x)
             sage: J = R.base_extend(QQ).ideal(2,x)
             sage: a = I.gen(); a
@@ -1395,8 +1396,9 @@ def gcd(self, other):
 
         ::
 
+            sage: R.<x> = ZZ[]
             sage: I = ZZ.ideal(7)
-            sage: J = ZZ[x].ideal(7,x)
+            sage: J = R.ideal(7,x)
             sage: I.gcd(J)
             Traceback (most recent call last):
             ...
@@ -1438,8 +1440,8 @@ def is_prime(self):
             False
             sage: ZZ.ideal(0).is_prime()
             True
-            sage: R.<x>=QQ[]
-            sage: P=R.ideal(x^2+1); P
+            sage: R.<x> = QQ[]
+            sage: P = R.ideal(x^2+1); P
             Principal ideal (x^2 + 1) of Univariate Polynomial Ring in x over Rational Field
             sage: P.is_prime()
             True

src/sage/rings/multi_power_series_ring_element.py

@@ -282,7 +282,7 @@ class MPowerSeries(PowerSeries):
 
     Convert elements from polynomial rings::
 
-        sage: R = PolynomialRing(ZZ,5,T.gens())
+        sage: R = PolynomialRing(ZZ,5,T.variable_names())
         sage: t = R.gens()
         sage: r = -t[2]*t[3] + t[3]^2 + t[4]^2
         sage: T(r)

src/sage/rings/polynomial/plural.pyx

@@ -734,7 +734,7 @@ cdef class NCPolynomialRing_plural(Ring):
                 return self._relations_commutative
 
             from sage.algebras.free_algebra import FreeAlgebra
-            A = FreeAlgebra( self.base_ring(), self.ngens(), self.gens() )
+            A = FreeAlgebra( self.base_ring(), self.ngens(), self.variable_names() )
 
             res = {}
             n = self.ngens()
@@ -748,7 +748,7 @@ cdef class NCPolynomialRing_plural(Ring):
             return self._relations
 
         from sage.algebras.free_algebra import FreeAlgebra
-        A = FreeAlgebra( self.base_ring(), self.ngens(), self.gens() )
+        A = FreeAlgebra( self.base_ring(), self.ngens(), self.variable_names() )
 
         res = {}
         n = self.ngens()

src/sage/rings/polynomial/polynomial_number_field.pyx

@@ -102,11 +102,12 @@ class Polynomial_absolute_number_field_dense(Polynomial_generic_dense_field):
 
         EXAMPLES::
 
-            sage: f = QQ[I][x].random_element()
+            sage: P.<x> = QQ[I][]
+            sage: f = P.random_element()
             sage: from sage.rings.polynomial.polynomial_number_field import Polynomial_absolute_number_field_dense
             sage: isinstance(f, Polynomial_absolute_number_field_dense)
             True
-            sage: a = QQ[I][x](x)
+            sage: a = P(x)
             sage: a.is_gen()
             True
         """

src/sage/rings/qqbar.py

@@ -4587,7 +4587,7 @@ def _more_precision(self):
         real which isn't the case without calling _ensure_real (see
         :trac:`11728`)::
 
-            sage: P = AA[x](1+x^4); a1,a2 = P.factor()[0][0],P.factor()[1][0]; a1*a2
+            sage: P = AA['x'](1+x^4); a1,a2 = P.factor()[0][0],P.factor()[1][0]; a1*a2
             x^4 + 1.000000000000000?
             sage: a1,a2
             (x^2 - 1.414213562373095?*x + 1, x^2 + 1.414213562373095?*x + 1)

src/sage/rings/quotient_ring.py

@@ -169,7 +169,9 @@ def QuotientRing(R, I, names=None):
     With polynomial rings (note that the variable name of the quotient
     ring can be specified as shown below)::
 
-        sage: R.<xx> = QuotientRing(QQ[x], QQ[x].ideal(x^2 + 1)); R
+        sage: P.<x> = QQ[]
+        sage: R.<xx> = QuotientRing(P, P.ideal(x^2 + 1))
+        sage: R
         Univariate Quotient Polynomial Ring in xx over Rational Field with modulus x^2 + 1
         sage: R.gens(); R.gen()
         (xx,)
@@ -860,7 +862,8 @@ def is_noetherian(self):
             sage: R.is_noetherian()
             True
 
-            sage: R = QuotientRing(QQ[x], x^2+1)
+            sage: P.<x> = QQ[]
+            sage: R = QuotientRing(P, x^2+1)
             sage: R.is_noetherian()
             True
 
@@ -898,7 +901,8 @@ def cover_ring(self):
 
         ::
 
-            sage: Q = QuotientRing(QQ[x], x^2 + 1)
+            sage: P.<x> = QQ[]
+            sage: Q = QuotientRing(P, x^2 + 1)
             sage: Q.cover_ring()
             Univariate Polynomial Ring in x over Rational Field
         """

src/sage/rings/real_arb.pyx

@@ -163,7 +163,7 @@ TESTS::
     Full MatrixSpace of 3 by 3 dense matrices over Real ball field
     with 53 bits precision
 
-    sage: polygen(RBF, x)^3
+    sage: polygen(RBF, 'x')^3
     x^3
 
 ::
@@ -374,7 +374,7 @@ class RealBallField(UniqueRepresentation, Field):
 
     ::
 
-        sage: (1/2*RBF(1)) + AA(sqrt(2)) - 1 + polygen(QQ, x)
+        sage: (1/2*RBF(1)) + AA(sqrt(2)) - 1 + polygen(QQ, 'x')
         x + [0.914213562373095 +/- 4.10e-16]
 
     TESTS::

src/sage/schemes/generic/algebraic_scheme.py

@@ -3339,8 +3339,8 @@ def dual(self):
         R = PolynomialRing(K, 'x', n + 1)
         Pd = sage.schemes.projective.projective_space.ProjectiveSpace(n, K, 'y')
         Rd = Pd.coordinate_ring()
-        x = R.gens()
-        y = Rd.gens()
+        x = R.variable_names()
+        y = Rd.variable_names()
         S = PolynomialRing(K, x + y + ('t',))
         if S.has_coerce_map_from(I.ring()):
             T = PolynomialRing(K, 'w', n + 1)
@@ -3789,7 +3789,7 @@ def segre_embedding(self, PP=None):
         #create new subscheme
         if PP is None:
             from sage.schemes.projective.projective_space import ProjectiveSpace
-            PS = ProjectiveSpace(self.base_ring(), M, R.gens()[AS.ngens():])
+            PS = ProjectiveSpace(self.base_ring(), M, R.variable_names()[AS.ngens():])
             Y = PS.subscheme(L)
         else:
             if PP.dimension_relative() != M:

src/sage/schemes/product_projective/space.py

@@ -1062,7 +1062,7 @@ def segre_embedding(self, PP=None, var='u'):
 
         #create new subscheme
         if PP is None:
-            PS = ProjectiveSpace(self.base_ring(), M, R.gens()[self.ngens():])
+            PS = ProjectiveSpace(self.base_ring(), M, R.variable_names()[self.ngens():])
             Y = PS.subscheme(L)
         else:
             if PP.dimension_relative() != M:

src/sage/schemes/projective/projective_homset.py

@@ -133,7 +133,7 @@ def points(self, B=0, prec=53):
                 N = PS.dimension_relative()
                 BR = X.base_ring()
                 #need a lexicographic ordering for elimination
-                R = PolynomialRing(BR, N + 1, PS.gens(), order='lex')
+                R = PolynomialRing(BR, N + 1, PS.variable_names(), order='lex')
                 I = R.ideal(X.defining_polynomials())
                 I0 = R.ideal(0)
                 #Determine the points through elimination

src/sage/schemes/projective/projective_morphism.py

@@ -1619,7 +1619,7 @@ def is_morphism(self):
             F.extend(defpolys)
             J = R.ideal(F)
         else:
-            S = PolynomialRing(R.base_ring().fraction_field(), R.gens(), R.ngens())
+            S = PolynomialRing(R.base_ring().fraction_field(), R.variable_names(), R.ngens())
             L = [S(f) for f in F] + [S(f) for f in defpolys]
             J = S.ideal(L)
         if J.dimension() > 0:
@@ -1798,7 +1798,7 @@ def primes_of_bad_reduction(self, check=True):
                 if R.base_ring().is_field():
                     J = R.ideal(F)
                 else:
-                    S = PolynomialRing(R.base_ring().fraction_field(), R.gens(), R.ngens())
+                    S = PolynomialRing(R.base_ring().fraction_field(), R.variable_names(), R.ngens())
                     J = S.ideal([S.coerce(F[i]) for i in range(R.ngens())])
                 if J.dimension() > 0:
                     raise TypeError("not a morphism")
@@ -1807,7 +1807,7 @@ def primes_of_bad_reduction(self, check=True):
 
                 #move the ideal to the ring of integers
                 if R.base_ring().is_field():
-                    S = PolynomialRing(R.base_ring().ring_of_integers(), R.gens(), R.ngens())
+                    S = PolynomialRing(R.base_ring().ring_of_integers(), R.variable_names(), R.ngens())
                     F = [F[i].change_ring(R.base_ring().ring_of_integers()) for i in range(len(F))]
                     J = S.ideal(F)
                 else:
@@ -1830,7 +1830,7 @@ def primes_of_bad_reduction(self, check=True):
                 if check:
                     index = 0
                     while index < len(badprimes):  #figure out which primes are really bad primes...
-                        S = PolynomialRing(GF(badprimes[index]), R.gens(), R.ngens())
+                        S = PolynomialRing(GF(badprimes[index]), R.variable_names(), R.ngens())
                         J = S.ideal([S.coerce(F[j]) for j in range(R.ngens())])
                         if J.dimension() == 0:
                             badprimes.pop(index)