On the existence of a point in the plane where repulsive central forces exerted by $ n $ fixed points cancel This is a physics-inspired question.
In what follows, $ \alpha \in (1,\infty) $ is a fixed constant, $ n \in \mathbb{N} $ a fixed integer $ \geq 2 $, and $ [n] \stackrel{\text{df}}{=} \mathbb{N}_{\leq n} $.

Let $ P_{1},\ldots,P_{n} $ be distinct points in the plane $ \mathbb{R}^{2} $. Let $ q_{1},\ldots,q_{n} $ be positive real numbers. Then for every $ i \in [n] $, define a vector-valued function $ \mathbf{F}_{i}: \mathbb{R}^{2} \setminus \{ P_{i} \}_{i \in [n]} \to \mathbb{R}^{2} $ by
$$
\forall X \in \mathbb{R}^{2} \setminus \{ P_{i} \}_{i \in [n]}: \quad
  {\mathbf{F}_{i}}(X) \stackrel{\text{df}}{=}
- \frac{q_{i}}{\left\| \overrightarrow{X P_{i}} \right\|^{\alpha}} \cdot
  \overrightarrow{X P_{i}}.
$$
(If $ \alpha = 3 $, then we can view $ q_{i} $ as a positive electrical charge carried by $ P_{i} $ and interpret $ {\mathbf{F}_{i}}(X) $ as a repulsive electrostatic force exerted on $ X $ by $ P_{i} $. Even if $ \alpha \in (1,3) \cup (3,\infty) $, we can still interpret $ {\mathbf{F}_{i}}(X) $ as some sort of repulsive central force exerted on $ X $ by $ P_{i} $.)

Question. Does there exist an $ X \in \mathbb{R}^{2} \setminus \{ P_{i} \}_{i \in [n]} $ such that $ \displaystyle \sum_{i \in [n]} {\mathbf{F}_{i}}(X) = \mathbf{0} $? In more physical terms, is there a point $ X $ at which the repulsive forces cancel? A rigorous argument is desired.

One thing is for sure. If such an $ X $ exists, then it must lie within the closed convex hull of $ \{ P_{i} \}_{i \in [n]} $.
Thank you for your help!
 A: I think there must be stationary points.
Consider a closed curve surrounding the vector field far from the sources of the vector field. The Index of the vector field around this curve must be $1$ (the vector field rotates one time counterclockwise along the curve). By Poincarè-Hopf theorem the indexes of the singular points inside the curve must sum up to +1 (the index along the curve) and since you have $n$ sources with indexes that sum up to a total of $+n$ to get $+1$ we need to have a $-n+1$ contribution i.e. at least another critical saddle point.
An online reference could be this one and also this from "Visual Complex Analysis" (starting from pag. 459).
A: A well-known special case too long for a comment:
If we consider point charges in the plane or line charges in space, the neatest choice of $\alpha$ is 2. In that case, if the $q_i$ are integers, you will find all stationary points of your force field as zeros of the derivative of the polynomial 
$$ p(z) \ := \ \prod_{i=1}^m (z - P_i)^{q_i} \, , $$
which are not zeros of $p(z)$ itself, called critical points of the second kind in the literature on geometry of polynomials. Here, $m$ is obviously the number of distinct zeros of $p(z)$. By clearing denominators you can obtain the critical points for rational charges in the same way and then consider limits for arbitrary positiv numbers.
The result that those critical points are contained in the convex hull of the zeros of $p(z)$ is called the Gauss - Lucas theorem in this context.
Such results are classical and can be found in any text on the geometry of polynomials like Morris Mardens book with that title from 1966, which gives a nice introduction and contains a lot of related material.
