For every integer N≥ 1, every collection of distinct points mathbf a_1,…,mathbf a_N∈ R^3, and every collection of nonzero charges q_1,…,q_N having the same sign, the set of equilibrium points mathbf x∈ R^3 setminusmathbf a_1,…,mathbf a_N:∑_i=1^N q_i fracmathbf x-mathbf a_i|mathbf x-mathbf a_i|^3=0 is finite.

Morse–Cairns finiteness conjecture for equilibria of Coulomb fields

The Morse–Cairns conjecture, posed by Morse and Cairns in 1969, asserts that point charges of one sign have only finitely many equilibrium points. A newly posted preprint claims this finiteness result and also gives bounds for mixed-sign charges.

Known results

Gabrielov, Novikov, and Shapiro (2007) obtained the first general, non-optimal upper bound for isolated equilibria.
A 2025 preprint gives, generically, the bound $2^{n}(3n-2)^{3}$ for $n$ charges in $\mathbb{R}^{3}$ .
Maxwell’s stronger proposed bound $(n-1)^{2}$ was claimed false in 2026: five positive charges allegedly have at least $24$ non-degenerate equilibria, without disproving finiteness.

September 2026 finiteness claim

The 2026 arXiv preprint claims to prove finiteness for one-sign charges and to bound mixed-sign equilibria away from the zero set of an auxiliary function. The result remains unverified in the retrieved evidence.

Current status (as of September 2026): Finiteness for one-sign charges is claimed proved, while independent verification is absent; Maxwell’s stronger bound is claimed false, and the mixed-sign question remains only partially bounded.

Morse–Cairns finiteness conjecture for equilibria of Coulomb fields

Primary source

Additional references

Progress summary

Solutions 0