more trivial cases

src/AlgebraicGeometry/Schemes/AffineSchemes/Objects/Methods.jl

@@ -187,7 +187,7 @@ Returns whether the scheme ``X`` is normal.
 
 # Examples
 ```jldoctest
-julia> R, (x, y, z) = rational_field()["x", "y", "z"];
+julia> R, (x, y, z) = QQ[:x, :y, :z];
 
 julia> X = spec(R);
 

src/AlgebraicGeometry/Schemes/CoveredSchemes/Objects/Constructors.jl

@@ -85,7 +85,7 @@ X_i \to X``, where ``X`` is the disjoint union of the covered schemes
 
 # Examples
 ```jldoctest
-julia> R_1, (x, y, z) = grade(rational_field()["x", "y", "z"][1]);
+julia> R_1, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]);
 
 julia> I_1 = ideal(R_1, z*x^2 + y^3);
 

src/AlgebraicGeometry/Schemes/CoveredSchemes/Objects/Methods.jl

@@ -168,7 +168,7 @@ Returns whether the scheme ``X`` is normal.
 
 # Examples
 ```jldoctest
-julia> R, (x, y, z) = rational_field()["x", "y", "z"];
+julia> R, (x, y, z) = QQ[:x, :y, :z];
 
 julia> X = covered_scheme(spec(R));
 
@@ -204,7 +204,7 @@ of non-integral schemes.
 
 # Examples
 ```jldoctest
-julia> R, (x, y, z) = grade(rational_field()["x", "y", "z"][1]);
+julia> R, (x, y, z) = graded_polynomial_ring(QQ, [:x, :y, :z]);
 
 julia> I = ideal(R, z*x^2 + y^3);
 

src/Rings/FreeAssociativeAlgebraIdeal.jl

@@ -99,7 +99,7 @@ Otherwise, returning `false` indicates an inconclusive answer, but larger `deg_b
 If `deg_bound` is not specified, the default value is `-1`, which means that no degree bound is imposed,
 resulting in a calculation using a much slower algorithm that may not terminate, but will return a full Groebner basis if it does.
 ```jldoctest
-julia> free, (x,y,z) = free_associative_algebra(QQ, ["x", "y", "z"]);
+julia> free, (x,y,z) = free_associative_algebra(QQ, [:x, :y, :z]);
 
 julia> f1 = x*y + y*z;
 
@@ -163,7 +163,7 @@ The default value of `deg_bound` is `-1`, which means that no degree bound is
 imposed, resulting in a computation that is usually slower, but will return a
 full Groebner basis if there exists a finite one.
 ```jldoctest
-julia> free, (x,y,z) = free_associative_algebra(QQ, ["x", "y", "z"]);
+julia> free, (x,y,z) = free_associative_algebra(QQ, [:x, :y, :z]);
 
 julia> f1 = x*y + y*z;
 

src/Rings/PBWAlgebra.jl

@@ -499,7 +499,7 @@ The generators of the returned algebra print according to the entries of `xs`. S
 
 # Examples
 ```jldoctest
-julia> D, (x, y, dx, dy) = weyl_algebra(QQ, ["x", "y"])
+julia> D, (x, y, dx, dy) = weyl_algebra(QQ, [:x, :y])
 (Weyl-algebra over Rational field in variables (x, y), PBWAlgElem{QQFieldElem, Singular.n_Q}[x, y, dx, dy])
 
 julia> dx*x
@@ -548,7 +548,7 @@ Return the opposite algebra of `A`.
 
 # Examples
 ```jldoctest
-julia> D, (x, y, dx, dy) = weyl_algebra(QQ, ["x", "y"])
+julia> D, (x, y, dx, dy) = weyl_algebra(QQ, [:x, :y])
 (Weyl-algebra over Rational field in variables (x, y), PBWAlgElem{QQFieldElem, Singular.n_Q}[x, y, dx, dy])
 
 julia> Dop, opp = opposite_algebra(D);
@@ -743,7 +743,7 @@ Return `true` if `I` is the zero ideal, `false` otherwise.
 
 # Examples
 ```jldoctest
-julia> D, (x, y, dx, dy) = weyl_algebra(QQ, ["x", "y"])
+julia> D, (x, y, dx, dy) = weyl_algebra(QQ, [:x, :y])
 (Weyl-algebra over Rational field in variables (x, y), PBWAlgElem{QQFieldElem, Singular.n_Q}[x, y, dx, dy])
 
 julia> I = left_ideal(D, [x, dx])
@@ -773,7 +773,7 @@ Return `true` if `I` is generated by `1`, `false` otherwise.
 
 # Examples
 ```jldoctest
-julia> D, (x, y, dx, dy) = weyl_algebra(QQ, ["x", "y"])
+julia> D, (x, y, dx, dy) = weyl_algebra(QQ, [:x, :y])
 (Weyl-algebra over Rational field in variables (x, y), PBWAlgElem{QQFieldElem, Singular.n_Q}[x, y, dx, dy])
 
 julia> I = left_ideal(D, [x, dx])
@@ -796,7 +796,7 @@ true
 ```
 
 ```jldoctest
-julia> D, (x, y, dx, dy) = weyl_algebra(GF(3), ["x", "y"]);
+julia> D, (x, y, dx, dy) = weyl_algebra(GF(3), [:x, :y]);
 
 julia> I = two_sided_ideal(D, [x^3])
 two_sided_ideal(x^3)
@@ -860,7 +860,7 @@ or `I` and `J` are a left and a right ideal, respectively, return the product of
 
 # Examples
 ```jldoctest
-julia> D, (x, y, dx, dy) = weyl_algebra(GF(3), ["x", "y"]);
+julia> D, (x, y, dx, dy) = weyl_algebra(GF(3), [:x, :y]);
 
 julia> I = left_ideal(D, [x^3+y^3, x*y^2])
 left_ideal(x^3 + y^3, x*y^2)
@@ -886,7 +886,7 @@ Given a two_sided ideal `I`, return the `k`-th power of `I`.
 
 # Examples
 ```jldoctest
-julia> D, (x, dx) = weyl_algebra(GF(3), ["x"]);
+julia> D, (x, dx) = weyl_algebra(GF(3), [:x]);
 
 julia> I = two_sided_ideal(D, [x^3])
 two_sided_ideal(x^3)
@@ -925,7 +925,7 @@ Return the intersection of two or more ideals.
 
 # Examples
 ```jldoctest
-julia> D, (x, y, dx, dy) = weyl_algebra(QQ, ["x", "y"]);
+julia> D, (x, y, dx, dy) = weyl_algebra(QQ, [:x, :y]);
 
 julia> I = intersect(left_ideal(D, [x^2, x*dy, dy^2])+left_ideal(D, [dx]), left_ideal(D, [dy^2-x^3+x]))
 left_ideal(-x^3 + dy^2 + x)
@@ -972,7 +972,7 @@ Return `true` if `f` is contained in `I`, `false` otherwise. Alternatively, use
 # Examples
 
 ```jldoctest
-julia> D, (x, dx) = weyl_algebra(QQ, ["x"]);
+julia> D, (x, dx) = weyl_algebra(QQ, [:x]);
 
 julia> I = left_ideal(D, [x*dx^4, x^3*dx^2])
 left_ideal(x*dx^4, x^3*dx^2)
@@ -982,7 +982,7 @@ true
 ```
 
 ```jldoctest
-julia> D, (x, y, dx, dy) = weyl_algebra(QQ, ["x", "y"]);
+julia> D, (x, y, dx, dy) = weyl_algebra(QQ, [:x, :y]);
 
 julia> I = two_sided_ideal(D, [x, dx])
 two_sided_ideal(x, dx)
@@ -1017,7 +1017,7 @@ end
 Return `true` if `I` is contained in `J`, `false` otherwise.
 # Examples
 ```jldoctest
-julia> D, (x, dx) = weyl_algebra(QQ, ["x"]);
+julia> D, (x, dx) = weyl_algebra(QQ, [:x]);
 
 julia> I = left_ideal(D, [dx^2])
 left_ideal(dx^2)
@@ -1052,7 +1052,7 @@ Return `true` if `I` is equal to `J`, `false` otherwise.
 
 # Examples
 ```jldoctest
-julia> D, (x, dx) = weyl_algebra(QQ, ["x"]);
+julia> D, (x, dx) = weyl_algebra(QQ, [:x]);
 
 julia> I = left_ideal(D, [dx^2])
 left_ideal(dx^2)

test/AlgebraicGeometry/Schemes/BlowupMorphism.jl

@@ -18,7 +18,7 @@
 end
 
 @testset "strict transforms of cartier divisors" begin
-  IP2 = projective_space(QQ, ["x", "y", "z"])
+  IP2 = projective_space(QQ, [:x, :y, :z])
   S = ambient_coordinate_ring(IP2)
   (x,y,z) = gens(S)
   I = ideal(S, [x, y])
@@ -37,7 +37,7 @@ end
 end
 
 @testset "isomorphism on complement of center" begin
-  P = projective_space(QQ, ["x", "y", "z"])
+  P = projective_space(QQ, [:x, :y, :z])
   S = homogeneous_coordinate_ring(P)
   (x, y, z) = gens(S)
   II = IdealSheaf(P, [x, y])

test/AlgebraicGeometry/Schemes/CartierDivisor.jl

@@ -63,7 +63,7 @@ end
 end
 
 @testset "conversion of Cartier to Weil divisors" begin
-  IP2 = projective_space(QQ, ["x", "y", "z"])
+  IP2 = projective_space(QQ, [:x, :y, :z])
   S = homogeneous_coordinate_ring(IP2)
   (x,y,z) = gens(S)
   I = ideal(S, x^2*y^3*(x+y+z))

test/AlgebraicGeometry/Schemes/CoherentSheaves.jl

@@ -44,7 +44,7 @@
 end
 
 @testset "Pushforward of modules" begin
-  IP = projective_space(QQ, ["x", "y", "z"])
+  IP = projective_space(QQ, [:x, :y, :z])
   S = homogeneous_coordinate_ring(IP)
   (x,y,z) = gens(S)
   f = z^2 - x*y
@@ -119,7 +119,7 @@ end
 end
 
 @testset "projectivization of vector bundles" begin
-  IP = projective_space(QQ, ["x", "y", "z", "w"])
+  IP = projective_space(QQ, [:x, :y, :z, :w])
   S = homogeneous_coordinate_ring(IP)
   (x,y,z,w) = gens(S)
   f = x^4 + y^4 + z^4 + w^4
@@ -155,7 +155,7 @@ end
 end
 
 @testset "direct sums of sheaves" begin
-  IP = projective_space(QQ, ["x", "y", "z", "w"])
+  IP = projective_space(QQ, [:x, :y, :z, :w])
   S = homogeneous_coordinate_ring(IP)
   (x, y, z, w) = gens(S)
   f = x^4 + y^4 + z^4 + w^4

test/AlgebraicGeometry/Schemes/CoveredScheme.jl

@@ -118,7 +118,7 @@
   end
 
   @testset "closed embeddings and singular loci" begin
-    IP2 = projective_space(QQ, ["x", "y", "z"])
+    IP2 = projective_space(QQ, [:x, :y, :z])
     S = homogeneous_coordinate_ring(IP2)
     (x, y, z) = gens(S)
     f = x^2*z + y^3 - y^2*z

test/AlgebraicGeometry/Schemes/FunctionFields.jl

@@ -119,7 +119,7 @@ end
 end
 
 @testset "pullbacks for function fields" begin
-  P = projective_space(QQ, ["x", "y", "z"])
+  P = projective_space(QQ, [:x, :y, :z])
   (x, y, z) = gens(homogeneous_coordinate_ring(P))
   Y = covered_scheme(P)
   II = ideal_sheaf(P, [x,y])
@@ -143,7 +143,7 @@ end
 end
 
 @testset "refinements" begin
-  P = projective_space(QQ, ["x", "y", "z"])
+  P = projective_space(QQ, [:x, :y, :z])
   S = homogeneous_coordinate_ring(P)
   (x, y, z) = gens(S)
   Y = covered_scheme(P)

test/AlgebraicGeometry/ToricVarieties/toric_schemes.jl

@@ -18,7 +18,7 @@
   end
 
   IP1 = projective_space(NormalToricVariety, 1)
-  set_coordinate_names(IP1, ["x", "y"])
+  set_coordinate_names(IP1, [:x, :y])
   Y = IP1*IP1
 
   @testset "Product of projective spaces" begin
@@ -27,7 +27,7 @@
   end
 
   IP2 = projective_space(NormalToricVariety, 2)
-  set_coordinate_names(IP2, ["x", "y", "z"])
+  set_coordinate_names(IP2, [:x, :y, :z])
   X, iso = Oscar.forget_toric_structure(IP2)
 
   @testset "Forget toric structure" begin

test/Rings/FreeAssociativeAlgebraIdeal.jl

@@ -1,6 +1,6 @@
 @testset "FreeAssociativeAlgebraIdeal.basic" begin
   Zt = polynomial_ring(ZZ, :t)[1]
-  R, (x, y, z) = free_associative_algebra(Zt, ["x", "y", "z", "w"])
+  R, (x, y, z) = free_associative_algebra(Zt, [:x, :y, :z, :w])
   I = ideal(R, [x*y*x, y*z^2])
   @test base_ring(I) == R
   for p in gens(R)
@@ -9,13 +9,13 @@
 end
 
 @testset "FreeAssociativeAlgebraIdeal.printing" begin
-  R, (x, y, z) = free_associative_algebra(GF(5), ["x", "y", "z", "w"])
+  R, (x, y, z) = free_associative_algebra(GF(5), [:x, :y, :z, :w])
   I = ideal(R, [x*y*x, y*z^2])
   @test length(string(I)) > 3
 end
 
 @testset "FreeAssociativeAlgebraIdeal.membership" begin
-  R, (x, y, z) = free_associative_algebra(QQ, ["x", "y", "z"])
+  R, (x, y, z) = free_associative_algebra(QQ, [:x, :y, :z])
   I = ideal(R, [x*y - y*x, x*z - z*x])
   @test !ideal_membership(x, I, 5)
   @test !ideal_membership(x, I, 10)
@@ -34,7 +34,7 @@ end
 end
 
 @testset "FreeAssociativeAlgebraIdeal.utils" begin
-  R, (x, y, z) = free_associative_algebra(QQ, ["x", "y", "z"])
+  R, (x, y, z) = free_associative_algebra(QQ, [:x, :y, :z])
   I = ideal(R, [x*y - y*x, x*z - z*x])
   @test base_ring(I) == R
   @test isa(ngens(I),Int)
@@ -46,15 +46,15 @@ end
   @test isa(lpring,NCRing)
 
   _, (x, y, z) = Singular.FreeAlgebra(QQ, ["x", "y","z"],6)
-  free, _ = free_associative_algebra(QQ, ["x", "y", "z"])
+  free, _ = free_associative_algebra(QQ, [:x, :y, :z])
   f1 = x*y + y*z
 
   F1 = free(f1)
   @test isa(F1,FreeAssociativeAlgebraElem)
 end 
 
 @testset "FreeAssociativeAlgebraIdeal.groebner_basis" begin
-    free, (x,y,z) = free_associative_algebra(QQ, ["x", "y", "z"])
+    free, (x,y,z) = free_associative_algebra(QQ, [:x, :y, :z])
     f1 = x*y + y*z
     f2 = x^2 + y^2
     I = ideal([f1, f2])

test/Rings/PBWAlgebra.jl

@@ -65,10 +65,10 @@ end
 end
 
 @testset "PBWAlgebra.weyl_algebra" begin
-  R, (x, dx) = weyl_algebra(QQ, ["x"])
+  R, (x, dx) = weyl_algebra(QQ, [:x])
   @test dx*x == 1 + x*dx
 
-  R, (x, y, dx, dy) = weyl_algebra(QQ, ["x", "y"])
+  R, (x, y, dx, dy) = weyl_algebra(QQ, [:x, :y])
   @test dx*x == 1 + x*dx
   @test dy*y == 1 + y*dy
   @test dx*y == y*dx
@@ -77,7 +77,7 @@ end
 end
 
 @testset "PBWAlgebra.opposite_algebra" begin
-  R, (x, y, dx, dy) = weyl_algebra(QQ, ["x", "y"])
+  R, (x, y, dx, dy) = weyl_algebra(QQ, [:x, :y])
   opR, M = opposite_algebra(R)
   @test M(dy*dx*x*y) == M(y)*M(x)*M(dx)*M(dy)
   @test inv(M)(M(x)) == x
@@ -91,7 +91,7 @@ end
 end
 
 @testset "PBWAlgebra.ideals" begin
-  R, (x, y, dx, dy) = weyl_algebra(QQ, ["x", "y"])
+  R, (x, y, dx, dy) = weyl_algebra(QQ, [:x, :y])
 
   I = left_ideal([x^2, y^2])
   @test length(string(I)) > 2
@@ -206,7 +206,7 @@ end
     @test eliminate(M.(I), M.([x, d])) == M.(left_ideal([a]))
   end
 
-  R, (x, dx) = weyl_algebra(QQ, ["x"])
+  R, (x, dx) = weyl_algebra(QQ, [:x])
   @test is_zero(eliminate(left_ideal([x*dx]), [x, dx]))
   @test is_one(eliminate(left_ideal([x, 1-x]), [x, dx]))