Is the solution of a polynomial an analytic function on the polynomial parameters?

Question

Be $\mu(z_1, \ldots, z_L)$ the only positive real solution to the equation

\begin{equation} \sum_{l=1}^L z_l \mu^l = 1 \end{equation}

With $z_1 = 1$, $z_l \geq 0 \forall l$. Clearly, varying the parameters $z_l$ varies the solution $\mu$. Now, is $\mu(z_1, \ldots, z_L)$ an analytic function? Can this be proved easily? Are there any references in literature I can read up?

I thought using the analytic implicit function theorem but I can't seem to find any reference suiting my case.