Linear bound for equilibria on lines and planes

Prove that the restriction of the electrostatic potential generated by n point charges in R^3 to any straight line or flat plane has at most O(n) equilibria.

Background

The paper proves an upper bound of p(n−1)+2n−1 for restrictions of V_p to lines when p is even. The authors state that they do not have a proof that this dependence on p can be replaced by a linear bound in n for arbitrary p, nor do they establish the corresponding planar result.

They therefore formulate a more restricted conjecture for the ordinary electrostatic potential V, asserting a linear bound for both one- and two-dimensional affine slices.

References

While we have no proof, we venture that this bound can be strengthened to at most $O(n)$ equilibria for any $p > 0$, and extended to $2$-dimensional slices. To focus, we formulate a more restricted version: \begin{conjecture} Let $V \colon R3 \setminus {A_1,A_2,\ldots,A_n} \to R$ be the electrostatic potential as in eqn:V. Then the restriction of $V$ to any straight line or flat plane in $R3$ has at most $O(n)$ equilibria. \end{conjecture}

— Counting Equilibria of the Electrostatic Potential  (2501.05315 - Edelsbrunner et al., 9 Jan 2025) in Conjecture slices, Section 4.3