The paper "On the finiteness issue of four-body balanced configurations in the plane" by Yuchen Wang and Lei Zhao (arXiv:2412.17639; published in Regular and Chaotic Dynamics 31(4):560-590, 2026) proves perturbatively that, for any four positive masses, the number of planar S\mathbf{S}-balanced configurations is finite whenever the anisotropy parameter η\eta of the matrix S\mathbf{S} is small enough, and states the conjecture that finiteness holds for generic positive masses for every η∈(0,1)\eta\in(0,1). The conjecture remains open: the only obstacle identified in the source is a class of "non-equal-order" singular diagrams that arise only for η>0\eta>0 and currently can be excluded only perturbatively. The problem posed here is to prove or refute that conjectured generic finiteness for arbitrary fixed η∈(0,1)\eta\in(0,1).

Background

The Newtonian NN-body problem describes NN point masses moving in the Euclidean plane under mutual gravitational attraction. A central configuration is a configuration of the bodies, with center of mass at the origin, such that the gravitational force on each body is proportional to its position vector: λqj=∑k≠jmk(qk−qj)/∥qk−qj∥3\lambda q_j=\sum_{k\neq j}m_k(q_k-q_j)/\|q_k-q_j\|^3. Planar central configurations generate the relative equilibria (solutions rotating rigidly at constant angular speed) and the homographic solutions (solutions in which the configuration keeps a fixed shape while rotating and dilating), and they govern total-collision and parabolic asymptotics of the dynamics. Balanced configurations, introduced by Albouy and Chenciner in their study of rigid motions of the NN-body problem in Euclidean space of arbitrary dimension, generalize central configurations: in the plane, a configuration solving λSqj=∑k≠jmk(qk−qj)/∥qk−qj∥3\lambda\mathbf{S}q_j=\sum_{k\neq j}m_k(q_k-q_j)/\|q_k-q_j\|^3 for some scalar λ\lambda and some positive-definite symmetric 2×22\times2 matrix S\mathbf{S} is called an S\mathbf{S}-balanced configuration (S\mathbf{S}BC); the case S=Id\mathbf{S}=\mathrm{Id} (a "numerical" matrix, i.e. a scalar multiple of the identity) recovers central configurations. Variationally, S\mathbf{S}BCs are the critical points of the Newtonian potential restricted to the weighted S\mathbf{S}-moment-of-inertia ellipsoid IS(q)=∑jmj Sqj⋅qj=1I_{\mathbf{S}}(\mathbf{q})=\sum_j m_j\,\mathbf{S}q_j\cdot q_j=1. For a non-numerical S\mathbf{S} (distinct eigenvalues) rotations no longer preserve the equation, so non-degenerate S\mathbf{S}BCs are isolated rather than appearing in rotation families. Since only the product λS\lambda\mathbf{S} enters, S\mathbf{S} is normalized by tr S=2\mathrm{tr}\,\mathbf{S}=2, and its eigenvalues are written 1+η1+\eta and 1−η1-\eta with 0≤η<10\le\eta<1; η\eta measures the anisotropy (eccentricity) of the inertia ellipsoid. Balanced configurations first appeared in Moeckel's work on clusters of small masses in central configurations, so their finiteness is closely tied to the classical finiteness question for central configurations. Whether, for fixed masses, there are only finitely many planar central configurations up to rotation and dilation is the finiteness conjecture of Wintner, listed by Smale among his mathematical problems for the twenty-first century. It was proved for four bodies by Hampton and Moeckel with a computer-assisted proof, and for five bodies outside a codimension-two algebraic set of masses by Albouy and Kaloshin. The Albouy-Kaloshin method algebraicizes the defining equations and argues by contradiction: an infinitude of configurations would produce a singular sequence along which some mutual distances tend to zero or infinity; the possible degeneration patterns are encoded in two-colored graphs ("singular diagrams"), each of which is either excluded outright or forces an explicit algebraic condition on the masses; a theorem of Chevalley on constructible sets (here: the image of an algebraic set under projection is a finite union of differences of algebraic sets) then yields finiteness for all masses outside the union of the exceptional algebraic sets — a statement "for generic masses", meaning for masses in a Zariski-open subset (the complement of a finite union of proper algebraic subsets) of the positive mass orthant. Wang and Zhao initiated the analogous finiteness question for balanced configurations. Their Theorem 1.1 shows that for any four positive masses there exists η0∈(0,1)\eta_0\in(0,1) such that finiteness holds for all 0<η<η00<\eta<\eta_0. The difficulty specific to η>0\eta>0 is that the algebraicized system loses the dilation symmetry that, for central configurations (η=0\eta=0), allows singular sequences to be normalized to "equal order"; as a result new "non-equal-order" singular diagrams appear which carry no mass condition and which the authors state "in general seem difficult to eliminate". They are excluded only for small η\eta (their Lemma 6.4). Conditionally on excluding non-equal-order sequences, their Proposition 6.3 reduces finiteness, for any fixed η\eta, to explicit codimension-one-or-two algebraic mass conditions attached to the equal-order diagrams I-VIII; the fully connected equal-order diagram IX is excluded not algebraically but by the fact that the S\mathbf{S}-moment of inertia ISI_{\mathbf{S}} cannot degenerate along a singular sequence (their Proposition 6.2). Their Theorem 8.1 additionally shows that perturbative finiteness extends to any number of bodies in the plane, conditional only on the corresponding central-configuration finiteness (itself still open for six or more bodies, and for five bodies outside the generic regime).

Problem Statement

Fix the planar Newtonian four-body problem with positive masses (m1,m2,m3,m4)(m_1,m_2,m_3,m_4) and center of mass at the origin, and fix a positive-definite symmetric 2×22\times2 matrix S\mathbf{S} with tr S=2\mathrm{tr}\,\mathbf{S}=2 and eigenvalues 1+η1+\eta and 1−η1-\eta for an arbitrary fixed η∈(0,1)\eta\in(0,1). Prove or refute the conjecture of Wang and Zhao, stated verbatim in their introduction: "We conjecture that finiteness holds true for generic choice of positive masses and 0<η<10<\eta<1." In the precise formulation adopted here — the reading consistent with the paper's Proposition 6.3, whose exceptional set Mη\mathcal{M}_\eta is allowed to depend on η\eta — the conjecture asserts that for every fixed η∈(0,1)\eta\in(0,1) there is a proper algebraic subset Eη⊊R4E_\eta\subsetneq\mathbb{R}^4 (possibly depending on η\eta) such that for all positive mass vectors outside EηE_\eta the system

λ Sqj=∑1≤k≤4,  k≠jmkqk−qj∥qk−qj∥3,j=1,…,4,\lambda\,\mathbf{S}q_j=\sum_{1\le k\le 4,\;k\neq j}m_k\frac{q_k-q_j}{\|q_k-q_j\|^3},\quad j=1,\dots,4,

in the unknowns (q1,q2,q3,q4)(q_1,q_2,q_3,q_4) and an auxiliary real scalar λ\lambda, has only finitely many solutions counted up to dilations qj↦cqjq_j\mapsto cq_j with c>0c>0. What is known is finiteness for any positive masses when η\eta is smaller than a mass-dependent threshold η0\eta_0 (Wang-Zhao, Theorem 1.1), and, for any fixed η\eta, finiteness off explicit codimension-1-or-2 mass conditions conditional on excluding non-equal-order singular sequences (Wang-Zhao, Proposition 6.3). The conjecture fails at a fixed η∈(0,1)\eta\in(0,1) exactly when infinitely many S\mathbf{S}-balanced configurations, counted up to dilation, occur on a set of positive masses that is not contained in any proper algebraic subset of R4\mathbb{R}^4; a single exceptional mass vector does not refute the generic statement, whereas failure at one fixed η\eta does.

References

  1. Yuchen Wang and Lei Zhao, "On the finiteness issue of four-body balanced configurations in the plane", arXiv:2412.17639 (2024), DOI 10.48550/arXiv.2412.17639, https://arxiv.org/abs/2412.17639
  2. Yuchen Wang and Lei Zhao, "On the Finiteness Issue of Four-Body Balanced Configurations in the Plane", Regular and Chaotic Dynamics 31(4):560-590 (2026), DOI 10.1134/S1560354726510052, https://doi.org/10.1134/S1560354726510052
  3. A. Albouy and A. Chenciner, "Le problème des n corps et les distances mutuelles", Inventiones Mathematicae 131(1):151-184 (1998), DOI 10.1007/s002220050200, https://doi.org/10.1007/s002220050200
  4. A. Albouy and V. Kaloshin, "Finiteness of central configurations of five bodies in the plane", Annals of Mathematics 176(1):535-588 (2012), DOI 10.4007/annals.2012.176.1.10, https://doi.org/10.4007/annals.2012.176.1.10
  5. M. Hampton and R. Moeckel, "Finiteness of relative equilibria of the four-body problem", Inventiones Mathematicae 163(2):289-312 (2006), DOI 10.1007/s00222-005-0461-0, https://doi.org/10.1007/s00222-005-0461-0
  6. R. Moeckel, "Relative equilibria with clusters of small masses", Journal of Dynamics and Differential Equations 9(4):507-533 (1997), DOI 10.1007/BF02219396, https://doi.org/10.1007/bf02219396
  7. L. Asselle and A. Portaluri, "Morse theory for S-balanced configurations in the Newtonian n-body problem", Journal of Dynamics and Differential Equations 35(1):907-946 (2023), DOI 10.1007/s10884-021-10036-y, https://doi.org/10.1007/s10884-021-10036-y
  8. S. Smale, "Mathematical problems for the next century", The Mathematical Intelligencer 20(2):7-15 (1998), DOI 10.1007/BF03025291, https://doi.org/10.1007/BF03025291
  9. K. Chang and K. Chen, "Toward finiteness of central configurations for the planar six-body problem by symbolic computations, (I) Determine diagrams and orders", Journal of Symbolic Computation 123:102277 (2024), DOI 10.1016/j.jsc.2023.102277, https://doi.org/10.1016/j.jsc.2023.102277
  10. Anders N. Jensen and Anton Leykin, "Smale's 6th problem for generic masses", arXiv:2301.02305 (2023), https://arxiv.org/abs/2301.02305
  11. Thiago Dias, "The Veronese Geometry of Dziobek Configurations and Generic Finiteness for Homogeneous Potentials", arXiv:2601.07962 (2026), https://arxiv.org/abs/2601.07962
  12. Katharina Kormanna and Giorgia Testolina, "Bifurcations of planar balanced configurations for the n-body problem in R4\mathbb{R}^4", arXiv:2510.17749 (2025), https://arxiv.org/abs/2510.17749
  13. Luca Asselle, Marco Fenucci and Alessandro Portaluri, "Bifurcations of balanced configurations for the Newtonian n-body problem in R4\mathbb{R}^4", arXiv:2011.09291 (2020), https://arxiv.org/abs/2011.09291