# diagrams of Riemann surfaces of multiple roots

I am stuck with an exercise that demands me to draw diagrams of Riemann surfaces of functions that consist of roots - and their compositions, e.g. $\sqrt{z}+\sqrt{z-1}$ or $\sqrt{1-\sqrt{z-1}}$.

I understand the idea, but are there any rules on how to work with roots of higher order or the sum of such roots? Because I didn't find any.

thank you very much,

Marie!

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I think the easiest way is to transform the problem into a polynomial equation, and then consider the algebraic variety that defines. For example, set $w = \sqrt{1 - \sqrt{z - 1}}$ and rearrange to get $(1 - w^2)^2 = z$. The Riemann surface in question is then (an open subset of) the solution locus of that equation. –  Zhen Lin Dec 2 '11 at 0:14