Finiteness of planar central configurations for six bodies with generic masses (Smale's 6th problem beyond $n=5$)

Whether the Newtonian $n$-body problem has only finitely many planar central configurations (relative equilibria) for generic positive masses — Smale's 6th problem for the 21st century — is settled for $n\leq 5$ (Albouy–Kaloshin for $n=5$) and open for $n\geq 6$. Jensen and Leykin re-proved the $n=5$ case by a tropical-geometric computation: for suitable valuations of the masses, every connected component of the tropical prevariety of the Albouy–Chenciner equations has a pointed recession cone, forcing the solution variety to be zero-dimensional. Their 2023/2025 paper leaves the method's core questions open: carrying the certification to $n=6$, where a 100-CPU-day computation produced components with non-pointed recession cones, and finding a systematic criterion for choosing mass valuations that guarantee pointedness. The problem posed here is to resolve generic finiteness for the planar six-body problem: whether the number of planar central configurations of six bodies with generic positive masses is finite. As of late 2025 the case remains open: the Chang–Chen symbolic-computation program has excluded 61 of at most 85 candidate 's-diagrams' only up to codimension-1 mass varieties, and no work has completed the tropical certification for $n=6$.

Background

A central configuration of the Newtonian $n$-body problem is a configuration of $n$ point masses that collapses homothetically to the center of mass under the Newtonian attraction, i.e. a shape for which the gravitational acceleration of each body is a common scalar multiple of its position relative to the center of mass. Central configurations are in one-to-one correspondence with relative equilibria — rigidly rotating solutions of the equations of motion — and their count controls bifurcations and compactifications of the $n$-body phase space. Smale's 6th problem (S. Smale, 'Mathematical problems for the next century', 1998) asks whether the number of relative equilibria of the planar $n$-body problem is finite for any choice of positive masses, with the generic-mass five-body case suggested as a first target; the generic-mass formulation — 'generic masses' means the masses avoid the zero set of some nonzero polynomial, i.e. fill a Zariski-open subset of the mass parameter space — is the strongest version that can hold, since Roberts (1999) exhibited a special (non-generic) set of five-body masses carrying a continuum of relative equilibria. Modulo rotation and scaling, central configurations are the solutions of the Albouy–Chenciner equations (Albouy–Chenciner 1997), a polynomial system in the $n(n-1)/2$ mutual distances made rotation-invariant by normalization; finiteness of central configurations for given masses is equivalent to this normalized solution set being zero-dimensional. The finiteness boundary is: $n=3$ classical (Euler and Lagrange), $n=4$ by Hampton–Moeckel (2006, Inventiones), $n=5$ planar with generic masses by Albouy–Kaloshin (2012, Annals of Mathematics) via an intricate algebraic elimination, and spatial $n=5$ by Hampton–Jensen (2011). Everything is open for $n\geq 6$. Jensen and Leykin ('Smale's 6th problem for generic masses', 2023/2025) gave an alternative, computation-driven proof for $n\leq 5$: work over the field of Puiseux series (formal power series with rational exponents), whose valuation map takes a series to its lowest exponent; tropical geometry replaces a variety over this field by a polyhedral 'shadow' (the tropical prevariety of its defining polynomials — the intersection of the polyhedral complexes dual to each polynomial's Newton polytope), and the equality of dimensions of a variety and its tropicalization, together with the fact that a balanced tropical curve (a weighted rational polyhedral complex satisfying the balancing condition at each cell) cannot lie in a polyhedron whose recession cone — the cone of directions in which the polyhedron is unbounded — is pointed (contained in an open half-space), imply that pointed recession cones of every connected component force the variety to be zero-dimensional. For $n=5$ and mass valuations $(1,4,9,16,25)$ (a computation of about 80 CPU-minutes in the software gfan, which computes Gröbner fans and tropical prevarieties), all 257 components were pointed, giving finiteness; the natural choice $(0,1,2,3,4)$ fails, while $(1,3,9,27,81)$ works even better. For $n=6$, a prevariety computation of about 100 CPU-days produced components with non-pointed recession cones, so the certification did not terminate; the authors state that more experimentation with valuations or with different versions of the Albouy–Chenciner equations is needed, and they give no criterion for which valuations can work. It remains open whether some mass-valuation vector $(v_1,\dots,v_6)\in\mathbb{Q}^6$ (or a variant version of the Albouy–Chenciner equations) makes every connected component of the six-body tropical prevariety have a pointed recession cone, and whether a systematic criterion exists for selecting such valuation vectors — explaining both the success of $(1,4,9,16,25)$-type valuations and the failure of $(0,1,2,3,4)$-type valuations; the published $n=5$ verification is packaged as a machine-replayable gfan recipe with exact, overflow-safe arithmetic, while no analogous certificate for $n=6$ has been produced. In parallel, Chang and Chen (2024, 2025) attack $n=6$ with symbolic computation along the Albouy–Kaloshin route: Part I determined 86 candidate 's-diagrams' (matrices associated with bicolored graphs encoding possible mechanisms of infinitude), and Part II reduced the list to at most 85 with 61 of them excluded except on codimension-1 varieties in the mass space, explicitly discussing the difficulties of the remaining cases. Generic finiteness has been proved only for a symmetric class of six-body configurations (Dias–Pan 2020), and computer-assisted exact counts exist for several exceptional five-body mass sets (Moczurad–Zgliczyński 2026).

Problem Statement

Resolve the next open case of Smale's 6th problem for generic masses in the planar Newtonian setting: prove that for $n=6$ bodies and generic positive masses (masses outside a proper algebraic subset of the mass space) the number of equivalence classes of planar central configurations — equivalently, solutions of the normalized Albouy–Chenciner distance equations — is finite; or refute this by exhibiting an explicit algebraic family of normalized central configurations valid for a Zariski-dense set of masses. The finiteness goal itself is to extend certified generic finiteness of planar central configurations from five to six bodies.

References

1. 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
2. A. Albouy and A. Chenciner, 'Le problème des n corps et les distances mutuelles', Inventiones Mathematicae 131(1):151–184 (1997), DOI 10.1007/s002220050200, https://doi.org/10.1007/s002220050200
3. 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
4. A. Albouy and V. Kaloshin, 'Finiteness of central configurations of five bodies in the plane', Annals of Mathematics (2) 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 A. Jensen, 'Finiteness of spatial central configurations in the five-body problem', Celestial Mechanics and Dynamical Astronomy 109(4):321–332 (2011), DOI 10.1007/s10569-010-9328-9, https://doi.org/10.1007/s10569-010-9328-9
6. G. E. Roberts, 'A continuum of relative equilibria in the five-body problem', Physica D: Nonlinear Phenomena 127(3–4):141–145 (1999), DOI 10.1016/S0167-2789(98)00315-7, https://doi.org/10.1016/S0167-2789(98)00315-7
7. A. N. Jensen and A. Leykin, 'Smale's 6th problem for generic masses', Journal of Experimental Mathematics 1(2):306–314 (2025), DOI 10.56994/JXM.001.002.006, arXiv:2301.02305, https://arxiv.org/abs/2301.02305
8. K.-M. Chang and K.-C. 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
9. K.-M. Chang and K.-C. Chen, 'Toward finiteness of central configurations for the planar six-body problem by symbolic computations. (II) Determine mass relations', SIAM Journal on Applied Dynamical Systems 24(3):2369–2404 (2025), DOI 10.1137/24M1716070, https://doi.org/10.1137/24M1716070
10. T. Dias and B.-Y. Pan, 'Generic finiteness for a class of symmetric planar central configurations of the six-body problem and the six-vortex problem', Journal of Dynamics and Differential Equations 32(4):1579–1602 (2020), DOI 10.1007/s10884-019-09765-y, https://doi.org/10.1007/s10884-019-09765-y
11. M. Moczurad and P. Zgliczyński, 'Central Configurations with Unequal Masses: Finiteness in Several Exceptional Cases of Five Bodies', arXiv:2601.01165 (2026), https://arxiv.org/abs/2601.01165
12. Y. Wang and L. 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, arXiv:2412.17639, https://arxiv.org/abs/2412.17639
13. A. N. Jensen, 'Gfan, a software system for Gröbner fans and tropical varieties', software, version 0.7, https://math.au.dk/~jensen/software/gfan/gfan.html