The finiteness conjecture for equilibria of electric fields generated by point charges of one sign

We prove that the electric field generated in three-dimensional space by finitely many point charges of one sign has only finitely many equilibrium points, thereby answering a 1969 question of Morse and Cairns (restated by Eremenko in 2008 and, as a conjecture, by Shapiro in 2015). More generally, for nonzero charges $q_i$ of possibly mixed signs at distinct sites $\mathbf a_i\in\mathbf{R}^3$, we show that the Coulomb field has at most $2^{N-4}(N-1)(9N^2+9N+10)$ equilibria in the region where $S(\mathbf x):=\sum_iq_i|\mathbf x-\mathbf a_i|^{-3}$ does not vanish. Of course, for charges of one sign, $S$ is nonzero everywhere. The proof rules out curves of equilibria using algebraic geometry and complex analysis on an associated complex curve, and then applies a Bézout count to obtain a quantitative bound. Well-known examples show that, in the mixed-sign case, the Coulomb field can vanish on curves contained in the zero set of $S$.