On well separated circular regions in the Riemann sphere and complex polynomials It started with a conjecture I had, see A statement on complex polynomials, which was false for $n \geq 3$, as shown by Noam D. Elkies in his answer there. The present post is an attempt to salvage the main idea of the conjecture by adding an extra hypothesis, namely that the $n$ circular disks are well separated. I will try to keep the remainder of this post self-contained, as much as possible.
Let $n \geq 3$ be an integer and let $D_i$, for $i = 1, \ldots, n$, be $n$ closed circular regions on the Riemann sphere $\widehat{\mathbb{C}}$, which are mutually disjoint. Note that for example a closed half-plane in the complex plane with $\infty$ is to be considered a closed circular region, since a line in the complex plane can be thought of as a circle on $\widehat{\mathbb{C}}$ passing through $\infty$.
The first problem I would like to solve, is how to define a measure, say $\delta$, of how well the $D_i$ are separated in such a way that $\delta$ is invariant under the group $PSL(2, \mathbb{C})$ of Moebius transformations.
Here is a suggestion. Given $i$, with $1 \leq i \leq n$, choose a Moebius transformation $\phi_i$ which maps $D_i$ onto the closed upper hemisphere of $\widehat{\mathbb{C}}$ (or, if you prefer, the complement of the open unit disk). Note that $\phi_i$ is not unique, as one could post-compose $\phi_i$ with an element of $PSU(1,1)$ (the group of Moebius transformations which preserve the open unit disk, isomorphic to the group of hyperbolic symmetries of the hyperbolic plane) and obtain another equally valid map.
Consider now the set $\mathcal{D}_i$ of circular regions $\phi_i(D_j)$, for $1 \leq j \leq n$ and $j \neq i$, all lying in the open unit disk.
Using the Poincare model on the open unit disk and the corresponding notion of hyperbolic distance, we let $\delta_i$ be the smallest pairwise hyperbolic distance between the elements of $\mathcal{D}_i$.
Finally, we let
$$ \delta = \operatorname{min}\{ \delta_i \,;\, 1 \leq i \leq n \}. $$
The number $\delta$ can be thought of as measuring how well the $D_i$ are separated. I believe, unless I am missing something, that $\delta$ is well defined and invariant under $PSL(2, \mathbb{C})$. If there are some issues with $\delta$, please inform me in the comments.
Let $p_i(z)$, for $1 \leq i \leq n$, be $n$ complex polynomials of degree at most $n - 1$ (so that it is allowed to have $\infty$ as a root, so to speak, when thinking about the Riemann sphere picture). We say that the $p_i(z)$ are compatible with the closed circular regions $D_i$ if, for each $i$, with $1 \leq i \leq n$, $p_i(z)$ has exactly $1$ root in each $D_j$, with $1 \leq j \leq n$ and $j \neq i$, and no other roots.
Given an integer $n \geq 3$, does there exist a constant $c_n > 0$, depending on $n$ only, such that if the $D_i$ ($1 \leq i \leq n$) are mutually disjoint closed circular regions in $\widehat{\mathbb{C}}$ with $\delta > c_n$ and if the $p_i(z)$ ($1 \leq i \leq n$) are complex polynomials of degree at most $n - 1$ which are compatible with the $D_i$, then the $p_i(z)$, for $1 \leq i \leq n$, are then linearly independent over $\mathbb{C}$?
I am also interested if one could find, assuming the above question has a positive answer, a specific value of $c_n$ for which the above statement is true (ideally the best value of $c_n$, though that may prove to be too difficult).
 A: Let $\{ z_1,\ldots,z_n\}$ be any finite set,
and
$$L_k(z)=\prod_{j\neq k}(z-z_j).$$
Then $L_k,\; 1\leq k\leq n$ are linearly independent since
the set of their linear combinations consists of all polynomials
of degree $\leq n-1$ (Lagrange interpolation formula says that),
and the space of all polynomials of degree $\leq n-1$ is of dimension $n$.
Now we can easily prove (by contradiction) that for every
$\{ z_1,\ldots,z_n\}$ there exists $\epsilon>0$ such that
the disks $\{ z:|z-z_k|<\epsilon\}$ have your desired property.
I understand that this is not what you desire; you want a
conformally invariant "separation parameter".
However this argument shows that if the "separation parameter" of a finite collection of disks has the property that when it tends to infinity, the disks shrink to points, then your statement about linear independence is true.
The separation parameter that you propose does not have this property, and in fact it has another important drawback: it can stay $>$ some positive number while some of your disks are arbitrarily close to the unit circle (which is the boundary of one of the disks of your family). And your separation parameter depends on
which disk you choose to send to the "upper hemisphere". So your
"separation parameter" can be very large, while some disks are very close to each other.
I can propose a separation parameter which will have the desired property: it is the minimum (over $1\leq k\leq n$) of the reciprocal extremal length of the family of curves
which separate one disk $D_k$ from the rest.
When this tends to $\infty$, this means that all these extremal lengths are small, so the disks are well separated, and must shrink to points.
For the definition and properties of extremal length, see any of the two books by Ahlfors, Conformal invariants, or Lectures on quasiconformal mappings.
