Re: Symmetrized complex space

Graven Water wrote:
A polynomial of degree n in C[x] has n roots counting multiplicities. The roots can be seen as elements of C^n/S, the quotient space of C^n by permutations of the coordinates - so that, for example, in C^2/S (a,b)=(b,a).

There's a map from the roots of polynomials in C^n/S to their coefficients
in C^n, which is injective by unique factorization, onto by the fundamental theorem of algebra, and it's continuous and has a continuous inverse. So it's a homeomorphism.

I think it's interesting that C^n folded in this strange way is actually manifold homeomorphic to C^n.

Laura


Almost (you have either to know about quotient spaces / invariants
or have to care for the holomorphic or algebraic functions on the
topological space, which are the those, which are invariant under
the action, ie. are their symmetrizations up to n!).

And that is no longer true for the multivariate situation.
.

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