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Suppose we have one polynom

##P(r_1, r_2, \ldots, r_n) = 0##

in n complex variables. This defines a n-1 dimensional complex algebraic surface.

Suppose that for each variable we have

##r_i = e^{ip_i}##

with complex p.

In the case n=1 of one variable r this results in the complex logarithm

##p = -i\ln r##

and we have to deal with a Riemann surface (but only for a discrete set of solutions of P)

Is there something like a Riemann surface (in more than one variable) over a (higher-dimensional) algebraic surface?

What can we say about the manifold defined as the solution of

##P\left( e^{ip_1}, e^{ip_2}, \ldots, e^{ip_n} \right) = 0##

in p-space?

##P(r_1, r_2, \ldots, r_n) = 0##

in n complex variables. This defines a n-1 dimensional complex algebraic surface.

Suppose that for each variable we have

##r_i = e^{ip_i}##

with complex p.

In the case n=1 of one variable r this results in the complex logarithm

##p = -i\ln r##

and we have to deal with a Riemann surface (but only for a discrete set of solutions of P)

**What happens in the case n>1?**Is there something like a Riemann surface (in more than one variable) over a (higher-dimensional) algebraic surface?

What can we say about the manifold defined as the solution of

##P\left( e^{ip_1}, e^{ip_2}, \ldots, e^{ip_n} \right) = 0##

in p-space?

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