Say, you have a function from $C^n \to C$ which is built by adding and composing polynomials and square root functions (e.g., $f(x_1,x_2,x_3)=\sqrt{x_1^2+1}+\sqrt{x_2}+2\sqrt{x_3}$). Is there an easy argument why the set of roots of such functions has measure 0 (provided that this is not the zero-function)? The argument is clear for polynomials, but what if we have these square roots? Any reference or pointers to related results are appreciated!

(a suitable squaring of the resulting equation leads to a polynomial system for this particular example, but this argument seems not to work in general...)

a priori. How do you deal with branch choices for the roots, for example? $\endgroup$