Suppose that $$x=u^5 - 10 u^3 v^2 + 5 u v^4$$ and $$y=5 u^4 v - 10 u^2 v^3 + v^5.$$ Given $x,y,u \in \mathbb{R}$, is it possible to find a $v \in \mathbb{R}$ that satisfies the above relations? I don't necessarily want an explicit form of the solution, but I need to know what conditions on $x$, $y$, or $u$ determine existence (and uniqueness, ideally). Either that, or if someone could point me towards a reference on the matter.
I have tried using "Reduce" and "Solve" with Mathematica without success (although I am still very new to the program).
This system of equations is from $z=x+iy=(u+iv)^5$. I am trying to determine if the surface $\{(x,y,u):u,v \in \mathbb{R}\}$ is a representation of the Riemann surface for $f(z)=z^{1/5}$. In order to do this, I believe it suffices to show that we can solve for $v$ given $x,y,u$, and that any non-unique solutions lie on the self-intersecting regions of the surface (where $y=0$). For an example with the cube root, see page 15 here.
$\dfrac{x}{u^5} = 1 - 10\left(\dfrac{v}{u}\right)^2 + 5\left(\dfrac{v}{u}\right)^4$,
and let $w = \left(\dfrac{v}{u}\right)^2 \Rightarrow \dfrac{x}{u^5} = 1 - 10w + 5w^2$ which is a quadratic equation in $w$.