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Germ surface of the square root of z to the eighth minus one

Over

D = C ∖ {z : z^{8} = 1}

, the two-valued square root is represented by

R = {(z, w) \in D \times C : w^{2} = z^{8} - 1} .

The maps

π (z, w) = z

and

E (z, w) = w

identify this unbranched double cover analytically with the corresponding component of the space of germs.

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Solution