Maxwell's problem for three point charges

Proofs that three point charges have at most four equilibria. (1) Charges of one sign: for every alpha > 0 the potential sum_i q_i |x - p_i|^(-2 alpha) in R^n has at most four critical points (sharp); the previous bound was six. In the angles under which the sides are seen from a critical point the equilibrium equations separate, the region of minima is convex, and the gradient of a strictly convex function is injective on it; no computer is used. (2) Charges of different signs: the Coulomb potential in R^3 has at most four critical points if the charges are not collinear, and at most four non-degenerate ones if they are; the same holds for every alpha in [1/4, 2]. One step, the convexity of an explicit planar region, is verified by interval arithmetic. Together this gives Maxwell's bound (k-1)^2 for k = 3 Coulomb charges: for all equilibria when the charges are not collinear, and for non-degenerate ones when they are. Includes the paper, the certificate code, numerical checks and internal audits.