Proving the non-degeneracy of the critical points of the potential function for a certain vector field with $ n $ point-singularities

Question

This question is an expansion of another question that I asked over at Math Stack Exchange.

In what follows, $ \alpha \in \mathbb{R}_{> 1} $ is a constant, $ n $ a fixed integer $ \geq 2 $, and $ [n] \stackrel{\text{df}}{=} \mathbb{N}_{\leq n} $.

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Let $ \mathbf{p}_{1},\ldots,\mathbf{p}_{n} $ be distinct points in $ \mathbb{R}^{2} $, and let $ q_{1},\ldots,q_{n} $ be positive real numbers. Then define a smooth vector field $ \mathbf{F}: \mathbb{R}^{2} \setminus \{ \mathbf{p}_{i} \}_{i \in [n]} \to \mathbb{R}^{2} $ by $$ \mathbf{F}(\mathbf{x}) \stackrel{\text{df}}{=} \sum_{i = 1}^{n} \frac{q_{i}}{\| \mathbf{x} - \mathbf{p}_{i} \|^{\alpha}} \cdot (\mathbf{x} - \mathbf{p}_{i}). $$ One can interpret $ \mathbf{F}(\mathbf{x}) $ as a sum of repulsive central forces exerted by $ \mathbf{p}_{1},\ldots,\mathbf{p}_{n} $ on $ \mathbf{x} $.

By a winding-number argument, one can prove there exists an $ \mathbf{x} \in \mathbb{R}^{2} \setminus \{ \mathbf{p}_{i} \}_{i \in [n]} $ lying inside the closed convex hull of $ \{ \mathbf{p}_{i} \}_{i \in [n]} $ such that $ \mathbf{F}(\mathbf{x}) = \mathbf{0} $. The argument runs roughly like this:

• Assume for the sake of contradiction that there is no such $ \mathbf{x} $.
• Then the only singularities of $ \mathbf{F} $ are $ \mathbf{p}_{1},\ldots,\mathbf{p}_{n} $.
• Let $ C $ be a circular counterclockwise contour that contains $ \mathbf{p}_{1},\ldots,\mathbf{p}_{n} $ in its interior.
• The winding number of $ \mathbf{F} $ with respect to $ C $ should be $ 1 $.
• Let $ \{ C_{i} \}_{i \in [n]} $ be a collection of circular counterclockwise contours such that for every $ i \in [n] $, (i) the center of $ C_{i} $ is $ \mathbf{p}_{i} $ and (ii) $ \mathbf{p}_{j} $ lies outside of $ C_{i} $ for every $ j \in [n] \setminus \{ i \} $.
• For every $ i \in [n] $, the winding number of $ \mathbf{F} $ with respect to $ C_{i} $ should also be $ 1 $.
• Hence, the total index of $ \mathbf{F} $ at its singularities is $ n $.
• However, this contradicts the Index Theorem (as $ n \neq 1 $).

If $ U $ denotes the potential function for $ \mathbf{F} $, then what this says is that $ U $ has a critical point.

Question. Is it true that every critical point of $ U $ is isolated, or even better, non-degenerate?

One can derive from the Hessian of $ U $ the system of equations that must hold for non-degeneracy to occur, but this system forms a wall that I am unable to surmount.

I would appreciate it if someone could offer suggestions on how to tackle this problem. References are also very welcome. Thank you!

Answer 1

I consider it highly unlikely that the critical points are always nondegenerate. Intuitively, you can imagine a finite number of such points for a generic choice of $q$, which could then "collide" for special choices of $q$. Even taking something as simple as regular $n$- -gon for $n>2$ and all $q=1$ might give a degenerate critical point at the origin (but I haven't done the calculation).

On the other hand, my intuition is that there should be no positive-dimensional sets of critical points, but I don't have a proof of this either.

Answer 2

My reason for this post is two-fold: (i) To give an answer for the case $ \alpha = 2 $. (ii) To attract attention to the other cases, which remain open.

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The potential function $ U: \mathbb{R}^{2} \setminus \{ \mathbf{p}_{i} \}_{i \in [n]} \to \mathbb{R} $ corresponding to $ \mathbf{F} $ is given by $$ \forall \mathbf{x} \in \mathbb{R}^{2} \setminus \{ \mathbf{p}_{i} \}_{i \in [n]}: \quad U(\mathbf{x}) = \sum_{i = 1}^{n} \frac{q_{i}}{2} \cdot \ln(\| \mathbf{x} - \mathbf{p}_{i} \|^{2}). $$ Let $ \mathbf{c} $ be a critical point of $ U $. Then let $ (l_{i})_{i \in [n]} $ be a sequence of parallel and non-intersecting rays in $ \mathbb{R}^{2} $ so that for every $ i \in [n] $, the following conditions hold:

• The starting point of $ l_{i} $ is $ \mathbf{p}_{i} $.
• $ \mathbf{c} \notin l_{i} $.

By these conditions, $ \displaystyle D \stackrel{\text{df}}{=} \mathbb{R}^{2} \bigg\backslash \bigcup_{i = 1}^{n} l_{i} $ is a connected open subset of $ \mathbb{R}^{2} $ that contains $ \mathbf{c} $.

Define a sequence $ (L_{i})_{i \in [n]} $ so that for every $ i \in [n] $, $ L_{i} $ denotes the complex logarithm function with branch cut $ l_{i} $. Identifying $ \mathbb{R}^{2} $ with $ \mathbb{C} $, the sum $ \displaystyle L \stackrel{\text{df}}{=} \sum_{i = 1}^{n} L_{i} $ is a non-constant holomorphic function on $ D $. Consequently, $ \mathbf{c} $ is an isolated critical point of $ U $ as $ U|_{D} = \Re(L) $.

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The question thus has an affirmative answer when $ \alpha = 2 $. However, the interesting case is $ \alpha = 3 $, which models inverse-square-law forces and so is more relevant to the natural world.