Monotonicity of the number of equal-mass central configurations with respect to the homogeneity exponent (Hampton's conjecture)

For the planar $n$-body problem with a homogeneous potential of degree $-\alpha$ (exponent $A=\alpha+2$; $A=3$ is the Newtonian case, $A=2$ the point-vortex case), it is conjectured — Hampton's Conjecture 6 (2019), restated as an open question in Natrup–Wang–Wang (2025) — that for equal masses the number of central configurations never decreases as the exponent $A$ increases, counting configurations up to rotation, scaling, and relabeling. A closely related open question asks whether, for fixed $n$, this count stabilizes for large $A$ away from the isolated bifurcation values where degenerate central configurations occur. Numerical evidence consistently shows counts increasing with $A$ through bifurcations that only create new classes, but no proof or counterexample is known.

Background

A central configuration (CC) of the planar $n$-body problem is a configuration of positions $q_1,\dots,q_n$ with masses $m_1,\dots,m_n$ such that each acceleration vector is a common scalar multiple of the position vector: $\sum_{j\ne i} m_j (q_j-q_i)/|q_j-q_i|^{A} = \lambda (q_i-q_0)$, where $q_0$ is the center of mass. Such configurations are exactly those that collapse homothetically to the center of mass and those that rotate rigidly as relative equilibria. For a homogeneous potential $U_\alpha=\sum_{i<j} m_i m_j |q_i-q_j|^{-\alpha}$ with $\alpha>0$ (and $\alpha=0$ the logarithmic/point-vortex limit, the singular-vortex interaction of ideal-fluid dynamics), the exponent is $A=\alpha+2$: $A=3$ gives the Newtonian potential and $A=2$ gives point vortices. CCs are counted up to the natural symmetries — rotation, dilation, and relabeling of equal masses — and, following the convention of the source papers, planar configurations are equivalent only under orientation-preserving (direct) isometries, so mirror-image configurations are counted as distinct. A CC is non-degenerate when the linearization of the CC equations at it is invertible (after factoring out the symmetry directions); degenerate CCs occur only at isolated values of $A$ for fixed masses, where bifurcations can change the count. By Morse theory — which relates the critical points of a smooth function to the topology of its domain — the CCs of a non-degenerate problem realize the topology of the shape sphere (complex projective space $\mathbb{CP}^{n-2}$), which yields the lower bound $l(n)=n!/2$ on the number of non-degenerate CCs for generic masses. Counting CCs is a classical hard problem: in the Newtonian case, finiteness for fixed masses is known only for four bodies in any dimension (Hampton–Moeckel 2006) and for five bodies in the plane (Albouy–Kaloshin 2012). Hampton (2019) studied how the number of equal-mass CCs depends on the exponent $A$, computing (for example) the equal-mass four-body Morse polynomial $6+24t+20t^2$ for all $A\ge 3$ (the Morse polynomial $\sum_k c_k t^k$ records, via $c_k$, the number of critical points of Morse index $k$, i.e. the dimension of the subspace on which the Hessian is negative definite) and observing numerically for $n=5$ that a bifurcation off the regular pentagon at $A\approx 6.755$ creates two new CC classes; in every computed case the count only increases with $A$. Natrup, Wang, and Wang (2025) recently proved that for any $\alpha\ge 0$ and nonzero masses the number of non-degenerate CCs is bounded above and below by constants depending only on $n$ (the upper bound from Khovanskii's fewnomial theorem, which bounds the number of nondegenerate real solutions of a polynomial system in terms of its number of distinct monomials; the lower from the Poincaré polynomial — the generating function of the Betti numbers, the dimensions of the homology groups — of $\mathbb{CP}^{n-2}$), and they highlight as open problems the sharpening of these qualitative bounds, the counting and classification of degenerate CCs for general $\alpha$ (in the Newtonian case Palmore 1975 exhibited degeneracies and Xia 1991 showed the set of masses admitting degenerate CCs has positive $(n-1)$-dimensional Hausdorff measure), and Hampton's monotonicity conjecture together with the stabilization question for large $A$.

Problem Statement

Prove or refute Hampton's conjecture (2019, Conjecture 6; recalled as open in Natrup–Wang–Wang 2025, Remark 2.3): for the planar $n$-body problem with equal masses and homogeneous potential exponent $A=\alpha+2\in[2,\infty)$ (i.e. $\alpha\ge 0$, including the point-vortex case $A=2$), the number of central configurations — counted up to rotation, dilation, and relabeling of the bodies, with the collinear and boundary strata of the shape sphere included and a fixed convention adopted at the isolated bifurcation values of $A$ where degenerate CCs occur — never decreases as $A$ increases, at regular (non-bifurcation) values of $A$. Include as part of the same question the stabilization corollary: whether for each fixed $n$ this count is eventually constant for all sufficiently large $A$ away from the isolated degenerate values, which would follow from monotonicity together with the uniform upper bound of Natrup–Wang–Wang. The relation is one-way: monotonicity implies the stabilization claim, but a refutation of monotonicity leaves stabilization open, so a resolution must settle stabilization unless it establishes monotonicity.

References

1. Julius Natrup, Qun Wang, Yuchen Wang, 'Finiteness of non-degenerate central configurations of the planar $n$-body problem with a homogeneous potential', arXiv:2502.20228 (2025), https://arxiv.org/abs/2502.20228
2. Marshall Hampton, 'Planar $N$-body central configurations with a homogeneous potential', Celestial Mechanics and Dynamical Astronomy 131, Paper No. 20 (2019), doi:10.1007/s10569-019-9898-0, arXiv:1810.13011, https://doi.org/10.1007/s10569-019-9898-0
3. Marshall Hampton, Richard Moeckel, 'Finiteness of relative equilibria of the four-body problem', Inventiones mathematicae 163 (2006), doi:10.1007/s00222-005-0461-0, https://doi.org/10.1007/s00222-005-0461-0
4. Alain Albouy, Vadim Kaloshin, 'Finiteness of central configurations of five bodies in the plane', Annals of Mathematics 176 (2012), doi:10.4007/annals.2012.176.1.10, https://doi.org/10.4007/annals.2012.176.1.10
5. Zhihong Xia, 'Central configurations with many small masses', Journal of Differential Equations 91 (1991), doi:10.1016/0022-0396(91)90137-X, https://doi.org/10.1016/0022-0396(91)90137-X
6. Julian I. Palmore, 'Classifying relative equilibria. II', Bulletin of the American Mathematical Society 81, 489-491 (1975), doi:10.1090/S0002-9904-1975-13794-3, https://doi.org/10.1090/S0002-9904-1975-13794-3
7. Thiago Dias, 'The Veronese Geometry of Dziobek Configurations and Generic Finiteness for Homogeneous Potentials', arXiv:2601.07962 (2026), https://arxiv.org/abs/2601.07962
8. Yiyang Deng, Jiangtao Xu, 'The Symmetry and Linear Stability of Convex 1+5 Coorbital Central Configurations with Homogeneous Potential', arXiv:2608.19958 (2026), https://arxiv.org/abs/2608.19958