adjust docu to writing :x instead of "x" etc. when constructing polyn… (

#4161)

* adjust docu to writing :x instead of "x" etc. when constructing polynomial rings

* independently, add one example to intersection theory

docs/src/CommutativeAlgebra/rings.md

@@ -27,7 +27,7 @@ The basic constructor below allows one to build multivariate polynomial rings:
 polynomial_ring(C::Ring, xs::AbstractVector{<:VarName}; cached::Bool = true)
 ```
 
-Given a ring `C` and a vector `xs` of  Strings, Symbols, or Characters, return 
+Given a ring `C` and a vector `xs` of  Symbols, Strings, or Characters, return 
 a tuple `R, vars`, say, which consists of a polynomial ring `R` with coefficient ring `C`
 and a vector `vars` of generators (variables) which print according to the entries of `xs`.
 
@@ -60,7 +60,7 @@ true
 ```jldoctest
 julia> R1, _ = polynomial_ring(ZZ, [:x, :y, :z]);
 
-julia> R2, _ = polynomial_ring(ZZ, [:x, :y, :z]);
+julia> R2, _ = polynomial_ring(ZZ, ["x", "y", "z"]);
 
 julia> R3, _ = polynomial_ring(ZZ, ['x', 'y', 'z']);
 

experimental/IntersectionTheory/src/Main.jl

@@ -1378,6 +1378,29 @@ basis(X::AbstractVariety, k::Int) = basis(X)[k+1]
 Return the Betti numbers of the Chow ring of `X`. Note that these are not
 necessarily equal to the usual Betti numbers, i.e., the dimensions of
 (co)homologies.
+
+# Examples
+```jldoctest
+julia> P2xP2 = abstract_projective_space(2, symbol = "k")*abstract_projective_space(2, symbol = "l")
+AbstractVariety of dim 4
+
+julia> betti(P2xP2)
+5-element Vector{Int64}:
+ 1
+ 2
+ 3
+ 2
+ 1
+
+julia> basis(P2xP2)
+5-element Vector{Vector{MPolyQuoRingElem}}:
+ [1]
+ [l, k]
+ [l^2, k*l, k^2]
+ [k*l^2, k^2*l]
+ [k^2*l^2]
+
+```
 """
 betti(X::AbstractVariety) = length.(basis(X))