Suppose $\vec{r}_1,\vec{r}_2,\cdots, \vec{r}_n \in \mathbb{R}^3$ and $Z_i \in \mathbb{Z}$ for each $i \in \{1,2,\cdots,n\}$. I would assume the electric/gravitational potential function $V : \mathbb{R}^3 \to \mathbb{R}$, $$ V(\vec{r}) = \sum_{j=1}^n \frac{Z_j}{\|\vec{r} - \vec{r}_j\|_2}; \qquad \| \ \cdot \ \|_2 \text{ euclidean norm}, $$ has been studied extensively by mathematicians in the past decades because of its use in natural sciences. I am interested in the zeros of the gradient $\nabla V$, but cannot find references about them. Ideally, I would like to (a) know how many there are for each choice of $n$ and collection $\{Z_1,\cdots,Z_n\}$, and/or (b) a formula for their coordinates. My question:
- Why am I not finding any references in my literary search? Is the problem too trivial to be studied, or is it simply unsolved? Or something else?