Prefer symbols over strings in polynomial_ring (#4133)

* Prefer symbols over strings in `polynomial_ring`

* Prefer symbols over strings in poly ring shorthand

docs/src/CommutativeAlgebra/GroebnerBases/groebner_bases.md

@@ -60,7 +60,7 @@ Here are some illustrating OSCAR examples:
 ##### Examples
 
 ```jldoctest
-julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
+julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> default_ordering(R)
@@ -90,7 +90,7 @@ Here are examples which indicate how to recover monomials, terms, and
 more from a given polynomial.
 
 ```jldoctest
-julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
+julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> f = 3*z^3+2*x*y+1
@@ -143,7 +143,7 @@ julia> tail(f)
 ```
 
 ```jldoctest
-julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"])
+julia> R, (x, y) = polynomial_ring(QQ, [:x, :y])
 (Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])
 
 julia> F = free_module(R, 3)

docs/src/CommutativeAlgebra/GroebnerBases/groebner_bases_integers.md

@@ -22,7 +22,7 @@ $f \in \mathbb Z[x]_>$, the notions *leading term*, *leading monomial*, *leading
 ##### Examples
 
 ```jldoctest
-julia> R, (x, y) = polynomial_ring(ZZ, ["x", "y"]);
+julia> R, (x, y) = polynomial_ring(ZZ, [:x, :y]);
 
 julia> reduce(3*x, [2*x])
 x
@@ -50,7 +50,7 @@ We refer to the  textbook [AL94](@cite) for more on this.
 ##### Examples
 
 ```jldoctest
-julia> R, (x,y) = polynomial_ring(ZZ, ["x","y"])
+julia> R, (x,y) = polynomial_ring(ZZ, [:x,:y])
 (Multivariate polynomial ring in 2 variables over ZZ, ZZMPolyRingElem[x, y])
 
 julia> I = ideal(R, [3*x^2*y+7*y, 4*x*y^2-5*x])

docs/src/CommutativeAlgebra/GroebnerBases/orderings.md

@@ -48,7 +48,7 @@ Here are some illustrating examples:
 ##### Examples
 
 ```jldoctest
-julia> S, (w, x) = polynomial_ring(QQ, ["w", "x"])
+julia> S, (w, x) = polynomial_ring(QQ, [:w, :x])
 (Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[w, x])
 
 julia> o = lex([w, x])
@@ -58,7 +58,7 @@ julia> canonical_matrix(o)
 [1   0]
 [0   1]
 
-julia> R, (w, x, y, z) = polynomial_ring(QQ, ["w", "x", "y", "z"])
+julia> R, (w, x, y, z) = polynomial_ring(QQ, [:w, :x, :y, :z])
 (Multivariate polynomial ring in 4 variables over QQ, QQMPolyRingElem[w, x, y, z])
 
 julia> o1 = degrevlex([w, x])
@@ -314,7 +314,7 @@ In OSCAR, block orderings are obtained by the concatenation of individual  order
 ##### Examples
 
 ```jldoctest
-julia> R, (w, x, y, z) = polynomial_ring(QQ, ["w", "x", "y", "z"])
+julia> R, (w, x, y, z) = polynomial_ring(QQ, [:w, :x, :y, :z])
 (Multivariate polynomial ring in 4 variables over QQ, QQMPolyRingElem[w, x, y, z])
 
 julia> o = degrevlex([w, x])*degrevlex([y, z])
@@ -405,7 +405,7 @@ basis vectors as *lex*, and to the $i > j$ ordering as *invlex*. And, we use the
 ##### Examples
 
 ```jldoctest
-julia> R, (w, x, y, z) = polynomial_ring(QQ, ["w", "x", "y", "z"]);
+julia> R, (w, x, y, z) = polynomial_ring(QQ, [:w, :x, :y, :z]);
 
 julia> F = free_module(R, 3)
 Free module of rank 3 over R

docs/src/CommutativeAlgebra/Miscellaneous/binomial_ideals.md

@@ -50,7 +50,7 @@ is_binomial(f::MPolyRingElem)
 is_binomial(I::MPolyIdeal)
 ```
 ```jldoctest
-julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
+julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> f = 2*x+y

docs/src/CommutativeAlgebra/ModulesOverMultivariateRings/complexes.md

@@ -31,7 +31,7 @@ Given a complex `C`,
 ##### Examples
 
 ```jldoctest
-julia> R, (x,) = polynomial_ring(QQ, ["x"]);
+julia> R, (x,) = polynomial_ring(QQ, [:x]);
 
 julia> F = free_module(R, 1);
 
@@ -86,7 +86,7 @@ with maps multiplied by $(-1)^d$.
 ##### Examples
 
 ```jldoctest
-julia> R, (x,) = polynomial_ring(QQ, ["x"]);
+julia> R, (x,) = polynomial_ring(QQ, [:x]);
 
 julia> F = free_module(R, 1);
 
@@ -150,7 +150,7 @@ is_exact(C::ComplexOfMorphisms{ModuleFP})
 ##### Examples
 
 ```jldoctest
-julia> R, (x,) = polynomial_ring(QQ, ["x"]);
+julia> R, (x,) = polynomial_ring(QQ, [:x]);
 
 julia> F = free_module(R, 1);
 
@@ -162,7 +162,7 @@ julia> a = hom(A, B, [x^2*B[1]]);
 
 julia> b = hom(B, B, [x^2*B[1]]);
 
-julia> R, (x,) = polynomial_ring(QQ, ["x"]);
+julia> R, (x,) = polynomial_ring(QQ, [:x]);
 
 julia> C = chain_complex([a, b]);
 

docs/src/CommutativeAlgebra/ModulesOverMultivariateRings/free_modules.md

@@ -67,7 +67,7 @@ If `F` is a free `R`-module, then
 ###### Examples
 
 ```jldoctest
-julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"]);
+julia> R, (x, y) = polynomial_ring(QQ, [:x, :y]);
 
 julia> F = free_module(R, 3);
 
@@ -114,7 +114,7 @@ Alternatively, directly write the element as a linear combination of basis vecto
 ##### Examples
 
 ```jldoctest
-julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"]);
+julia> R, (x, y) = polynomial_ring(QQ, [:x, :y]);
 
 julia> F = free_module(R, 3);
 
@@ -138,7 +138,7 @@ Given an element `f`  of a free module `F` over a multivariate polynomial ring w
 ##### Examples
 
 ```jldoctest
-julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"]);
+julia> R, (x, y) = polynomial_ring(QQ, [:x, :y]);
 
 julia> F = free_module(R, 3);
 

docs/src/CommutativeAlgebra/ModulesOverMultivariateRings/subquotients.md

@@ -80,7 +80,7 @@ If `M` is a subquotient with ambient free `R`-module `F`, then
 ##### Examples
 
 ```jldoctest
-julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
+julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> F = free_module(R, 1)
@@ -175,7 +175,7 @@ Alternatively, directly write the element as an $R$-linear combination of genera
 ##### Examples
 
 ```jldoctest
-julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
+julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> F = free_module(R, 1)
@@ -231,7 +231,7 @@ If this is already clear, it may be convenient to omit the test (`check = false`
 ##### Examples
 
 ```jldoctest
-julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
+julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> F = free_module(R, 1)

docs/src/CommutativeAlgebra/affine_algebras.md

@@ -67,7 +67,7 @@ If `A=R/I` is the quotient of a multivariate polynomial ring `R` modulo an ideal
 ###### Examples
 
 ```jldoctest
-julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"]);
+julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z]);
 
 julia> A, _ = quo(R, ideal(R, [y-x^2, z-x^3]));
 
@@ -141,7 +141,7 @@ or by directly coercing elements of `R` into `A`.
 ###### Examples
 
 ```jldoctest
-julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"]);
+julia> R, (x, y) = polynomial_ring(QQ, [:x, :y]);
 
 julia> A, p = quo(R, ideal(R, [x^3*y^2-y^3*x^2, x*y^4-x*y^2]));
 
@@ -225,7 +225,7 @@ If `a` is an ideal of the affine algebra `A`, then
 ###### Examples
 
 ```jldoctest
-julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"]);
+julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z]);
 
 julia> A, _ = quo(R, ideal(R, [y-x^2, z-x^3]));
 
@@ -359,9 +359,9 @@ kernel(F::AffAlgHom)
 ###### Examples
 
 ```jldoctest
-julia> D1, (w, x, y, z) = graded_polynomial_ring(QQ, ["w", "x", "y", "z"]);
+julia> D1, (w, x, y, z) = graded_polynomial_ring(QQ, [:w, :x, :y, :z]);
 
-julia> C1, (s,t) = graded_polynomial_ring(QQ, ["s", "t"]);
+julia> C1, (s,t) = graded_polynomial_ring(QQ, [:s, :t]);
 
 julia> V1 = [s^3, s^2*t, s*t^2, t^3];
 
@@ -383,7 +383,7 @@ Ideal generated by
 
 julia> C2, p2 = quo(D1, twistedCubic);
 
-julia> D2, (a, b, c) = graded_polynomial_ring(QQ, ["a", "b", "c"]);
+julia> D2, (a, b, c) = graded_polynomial_ring(QQ, [:a, :b, :c]);
 
 julia> V2 = [p2(w-y), p2(x), p2(z)];
 
@@ -403,9 +403,9 @@ Ideal generated by
 ```
 
 ```jldoctest
-julia> D3,y = polynomial_ring(QQ, "y" => 1:3);
+julia> D3,y = polynomial_ring(QQ, :y => 1:3);
 
-julia> C3, x = polynomial_ring(QQ, "x" => 1:3);
+julia> C3, x = polynomial_ring(QQ, :x => 1:3);
 
 julia> V3 = [x[1]*x[2], x[1]*x[3], x[2]*x[3]];
 
@@ -441,9 +441,9 @@ is_finite(F::AffAlgHom)
 ###### Examples
 
 ```jldoctest
-julia> D, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"]);
+julia> D, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z]);
 
-julia> S, (a, b, c) = polynomial_ring(QQ, ["a", "b", "c"]);
+julia> S, (a, b, c) = polynomial_ring(QQ, [:a, :b, :c]);
 
 julia> C, p = quo(S, ideal(S, [c-b^3]));
 
@@ -471,9 +471,9 @@ true
 ```
 
 ```jldoctest
-julia> R, (x, y, z) = polynomial_ring(QQ, [ "x", "y", "z"]);
+julia> R, (x, y, z) = polynomial_ring(QQ, [ :x, :y, :z]);
 
-julia> C, (s, t) = polynomial_ring(QQ, ["s", "t"]);
+julia> C, (s, t) = polynomial_ring(QQ, [:s, :t]);
 
 julia> V = [s*t, t, s^2];
 
@@ -528,7 +528,7 @@ noether_normalization(A::MPolyQuoRing)
 ###### Examples
 
 ```jldoctest; setup = :(Singular.call_interpreter("""system("random", 47);"""))
-julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"]);
+julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z]);
 
 julia> A, _ = quo(R, ideal(R, [x*y, x*z]));
 

docs/src/CommutativeAlgebra/ideals.md

@@ -30,7 +30,7 @@ If `I` is an ideal of a multivariate polynomial ring  `R`, then
 ###### Examples
 
 ```jldoctest
-julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"])
+julia> R, (x, y) = polynomial_ring(QQ, [:x, :y])
 (Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])
 
 julia> I = ideal(R, [x, y])^2
@@ -267,7 +267,7 @@ dehomogenizer(H::Homogenizer)
 ```
 
 ```jldoctest
-julia> P, (x, y) = polynomial_ring(QQ, ["x", "y"]);
+julia> P, (x, y) = polynomial_ring(QQ, [:x, :y]);
 
 julia> I = ideal([x^2+y, x*y+y^2]);
 

docs/src/CommutativeAlgebra/localizations.md

@@ -106,7 +106,7 @@ This reflects the way of creating localizations of quotients of multivariate pol
 ##### Examples
 
 ```jldoctest
-julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"]);
+julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z]);
 
 julia> P = ideal(R, [x])
 Ideal generated by
@@ -127,11 +127,11 @@ true
 ```
 
 ```jldoctest
-julia> T, t = polynomial_ring(QQ, "t");
+julia> T, t = polynomial_ring(QQ, :t);
 
 julia> K, a =  number_field(2*t^2-1, "a");
 
-julia> R, (x, y) = polynomial_ring(K, ["x", "y"]);
+julia> R, (x, y) = polynomial_ring(K, [:x, :y]);
 
 julia> I = ideal(R, [2*x^2-y^3, 2*x^2-y^5])
 Ideal generated by
@@ -180,7 +180,7 @@ multiplicatively closed subset of `R`, and `RQL` is the localization of `RQ` at
 ##### Examples
 
 ```jldoctest
-julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
+julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z])
 (Multivariate polynomial ring in 3 variables over QQ, QQMPolyRingElem[x, y, z])
 
 julia> P = ideal(R, [x])
@@ -222,11 +222,11 @@ y^2/z^2
 ```
 
 ```jldoctest
-julia> T, t = polynomial_ring(QQ, "t");
+julia> T, t = polynomial_ring(QQ, :t);
 
 julia> K, a =  number_field(2*t^2-1, "a");
 
-julia> R, (x, y) = polynomial_ring(K, ["x", "y"]);
+julia> R, (x, y) = polynomial_ring(K, [:x, :y]);
 
 julia> I = ideal(R, [2*x^2-y^3, 2*x^2-y^5])
 Ideal generated by
@@ -294,7 +294,7 @@ of representing `f` by pairs of elements of `RQ` and not the internal representa
 ##### Examples
 
 ```jldoctest
-julia> R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"]);
+julia> R, (x, y, z) = polynomial_ring(QQ, [:x, :y, :z]);
 
 julia> P = ideal(R, [x])
 Ideal generated by
@@ -330,11 +330,11 @@ true
 ```
 
 ```jldoctest
-julia> T, t = polynomial_ring(QQ, "t");
+julia> T, t = polynomial_ring(QQ, :t);
 
 julia> K, a =  number_field(2*t^2-1, "a");
 
-julia> R, (x, y) = polynomial_ring(K, ["x", "y"]);
+julia> R, (x, y) = polynomial_ring(K, [:x, :y]);
 
 julia> I = ideal(R, [2*x^2-y^3, 2*x^2-y^5])
 Ideal generated by
@@ -439,7 +439,7 @@ multivariate polynomial rings is similar..
 ##### Examples
 
 ```jldoctest
-julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"]);
+julia> R, (x, y) = polynomial_ring(QQ, [:x, :y]);
 
 julia> f = x^3+y^4
 x^3 + y^4