A polynomial [itex]p(z)=z^n+a_{n-1}z^{n-1}+...+a_0[/itex] has n roots [itex]\lambda_1,...,\lambda_n[/itex], and there's a map from the coefficients [itex](a_0,...,a_{n-1})\in C^n[/itex] to [itex](\lambda_1,...,\lambda_n)\in C^n/S_n[/itex], where [itex]S_n[/itex] is the symmetry group on n elements, and [itex]C^n/S_n[/itex] is complex n-space quotiented by permutations on the elements (since it doesn't matter what order the roots are in). [itex]C^n/S_n[/itex] has the quotient topology. This map [itex]C^n\rightarrow C^n/S_n[/itex] is injective because of unique factorization, surjective, and continuous, and it has a continuous inverse.
Does that mean that [itex]C^n[/itex] is homeomorphic to [itex]C^n/S_n?[/itex] That seems remarkable.
Does anybody recognize this space [itex]C^n/S_n[/itex], or know how to find out more about it?
Laura
Does that mean that [itex]C^n[/itex] is homeomorphic to [itex]C^n/S_n?[/itex] That seems remarkable.
Does anybody recognize this space [itex]C^n/S_n[/itex], or know how to find out more about it?
Laura