Moving from polynomials into radical functions: does algebraic geometry still work?

Question

Say I have a polynomial: $x + y = 0$, but subject to the constraints $x = \sqrt{1-u^2}$, $y = \sqrt{1-w^2}$, $u\in[-1,1],w\in[-1,1]$. I can re-write my original polynomial as an inequality: $\sqrt{1-u^2} + \sqrt{1-w^2} = 0, u\in[-1,1],w\in[-1,1]$, but then I've moved from polynomials into radical functions. Do the tools of algebraic geometry still work? In particular, can I still assume that the solution to the second formulation of the problem can be found as the union of solutions to a set of semi-algebraic sets?