How would you characterise the inverse image of this polynomial equation?

Question

I am postgraduate student, and in my project I am working with a function in several variables. I would be surprised if it has not been studied extensively, but as I am not a mathematician, I would like to consult with you to find out how I can learn more about it.

Suppose $\vec{y}_1,\vec{y}_2,\cdots,\vec{y}_m \in \mathbb{R}^d$ are fixed, and $a_i,b_j > 0$ for all $i,j$ then the function is $f:\mathbb{R}^{n \cdot d} \to \mathbb{R}$ given by $$ f(\vec{x}_1,\vec{x}_2\cdots,\vec{x}_n) = \sum_{i=1}^n \sum_{j=1}^m \frac{a_j}{|\vec{x}_i - \vec{y}_j|} + \sum_{i\neq j} \frac{b_j}{|\vec{x}_i - \vec{x}_j |}. $$ Here $|\cdot|$ is the Euclidean norm. For a given collection $\{\vec{y}_1,\vec{y}_2,\cdots,\vec{y}_m\}$, I want to find out as much as possible about the inverse images $f^{-1}(c)$ for each real number $c$.

I am not very experienced, so my question is basically this: How would you search for information for this problem? 1. In which field does this question belong? 2. What would you call an equation of this form?

Answer 1

Do you know about Morse functions? One can say that $f^{-1}(c)$ is a smooth manifold for all but finitely many values of $c$. If you vary $c$ the $f^{-1}(c)$ will by homotopy equivalent as long as you don't hit any of the points where it is not a smooth manifold. At these values, the preimage will have a singularity and the topology of the preimage will change.

In a simple example with say $d=2$ and $n=1$ one can see what is going on, the general picture is similar but hard to visualize due to higher dimensions.

As suggested by william in comments, the book "Morse Theory" by John Milnor is considered a good source and introduction to the area.

Answer 2

I would like to contribute to the intuitive understanding of the issue, probably not for you MikkelRev, but for people reading this question desiring to grasp the big figure.

Here is an example in the case where $d=n=2$ with $m=3$ "poles".

Let us give a graphical representation of function $f$ as a surface as shown below :

I have represented some level lines, in particular at the two saddle points where they take a "lemniscate" form, indicating a singularity, i.e., a topological change.

Generally speaking, above the highest saddle point, the $f^{-1}(c)$ can be described as the distinct "onions peels" that one gets as sections of each "chimney" at various heights by a horizontal plane. This plane can be imagined as an adjustable sea level which can be so high that all islands remain with a single peak.