A ‘strong’ form of the Fundamental Theorem of Algebra Let $ n \in \mathbb{N} $ and $ a_{0},\ldots,a_{n-1} \in \mathbb{C} $ be constants. By the Fundamental Theorem of Algebra, the polynomial
$$
p(z) := z^{n} + \sum_{k=0}^{n-1} a_{k} z^{k} \in \mathbb{C}[z]
$$
has $ n $ roots, including multiplicity. If we vary the values of $ a_{0},\ldots,a_{n-1} $, the roots will obviously change, so it seems natural to ask the following question.

Do the $ n $ roots of $ p(z) $ depend on the coefficients in an analytic sort of way? More precisely, can we find holomorphic functions $ r_{1},\ldots,r_{n}: \mathbb{C}^{n} \to \mathbb{C} $ such that
  $$
z^{n} + \sum_{k=0}^{n-1} a_{k} z^{k} = \prod_{j=1}^{n} [z - {r_{j}}(a_{0},\ldots,a_{n-1})]?
$$

The definition of a holomorphic function of several complex variables is given as follows:

Definition Let $ n \in \mathbb{N} $ and $ \Omega \subseteq \mathbb{C}^{n} $ be a domain (i.e., a connected open subset). A function $ f: \Omega \to \mathbb{C} $ is said to be holomorphic if and only if it is holomorphic in the usual sense in each of its $ n $ variables.

The existence of $ r_{1},\ldots,r_{n}: \mathbb{C}^{n} \to \mathbb{C} $ that are continuous seems to be a well-known result (due to Ostrowski, perhaps?), but I am unable to find anything in the literature that is concerned with the holomorphicity of these functions.
Any help would be greatly appreciated. Thank you very much! 

Such holomorphic functions exist locally as long as you stay away from places where two or more of the roots coincide, but the global situation is more complicated.  To get an idea of what happens, consider the case $n=2$, i.e., quadratic equations, and, to make things clearer yet, consider the subspace of $\mathbb C^2$ where $a_1=0$.  So you're looking at the equation $z^2+a_0=0$.  The solutions, of course, are the two branches of $\sqrt{-a_0}$.  They're locally holomorphic away from the origin, but there's a branch point at the origin, and the two solutions (away from the origin) are actually branches of a single, two-valued analytic function.  
The general situation is similar, but bigger and in higher dimensions.  Away from the locus $D$ where two (or more) of the roots coincide, you have $n$ local holomorphic functions, but these are branches of a single $n$-valued analytic function, with branching along the locus $D$.
