Sharp bounds and exact generic counts for Dziobek central configurations under homogeneous potentials

For the n-body problem with a homogeneous interaction potential, central configurations of Dziobek type (configurations whose n bodies affinely span dimension n-2, such as the planar four-body and spatial five-body cases) are known to be finite in number for generic masses, and Dias (2026) proved this simultaneously for the whole one-parameter family of homogeneous potentials (all exponents except the degenerate harmonic case $a=0$ , where every configuration is central), with a uniform Bezout-type upper bound (a product-of-degrees count from algebraic geometry) $2^{\binom{n+1}{2}+n-1}$ (8192 for n=4) independent of the potential exponent. The open problem is the quantitative core left by that theorem: determine the exact number of Dziobek central-configuration classes for generic masses (beginning with the n=4 case; the analogous count for general n at least 4 is the eventual goal of the program, not part of this question) and decide whether, and by how much, the uniform bound can be improved. No exact generic count is known in any non-symmetric case for any n at least 4.

Background

In the n-body problem with masses $m_1,\ldots,m_n$ at positions $x_1,\ldots,x_n\in\mathbb{R}^d$ , a central configuration is a configuration for which the total force on each body is proportional to its displacement from the center of mass $c$ : there is a constant $\lambda>0$ with $\sum_{i\neq j} m_i (x_i-x_j) r_{ij}^{2a}+\lambda (x_j-c)=0$ for every $j$ , where $r_{ij}$ is the mutual distance between bodies $i$ and $j$ . Central configurations generate the only explicitly known solutions of the n-body problem (homothetic collapse/expansion and rigidly rotating relative equilibria), govern the asymptotics of total collisions, and organize the bifurcation structure of the energy-momentum integral manifolds, so they are customarily counted up to translation, rotation, and dilation.

The source paper (Dias, January 2026) works with the one-parameter family of homogeneous potentials $U_a=\frac{1}{2a+2}\sum_{i<j}m_i m_j r_{ij}^{2a+2}$ for $a\neq -1$ (with $\sum_{i<j} m_i m_j \log r_{ij}$ replacing it at $a=-1$ ): $a=-3/2$ is the Newtonian case and, in the plane, $a=-1$ is the Helmholtz point-vortex problem, so one formalism covers both physical systems. Whether, for fixed positive masses, the number of central-configuration classes is finite is a classical question (raised in 1918 and popularized as the sixth of Smale's problems for the twenty-first century). Finiteness is known for n=3 (five classes, Euler and Lagrange), for collinear configurations of any n, for configurations spanning dimension n-1, for the four-body problem in all dimensions (Hampton and Moeckel, 2006, via computer-algebraic Gröbner-basis and resolution methods), for the planar five-body problem outside a codimension-two exceptional set in mass space (Albouy and Kaloshin, 2012), and generically for spatial five-body configurations (Hampton and Jensen, 2011); the general case remains open, which is why attention has shifted to generic finiteness — finiteness for all mass vectors outside a proper algebraic subvariety of mass space. Dziobek configurations are the central configurations whose affine hull has dimension exactly n-2 (planar for n=4, spatial for n=5). Dziobek (1900) introduced the mutual distances as primary variables and the equations now bearing his name, which in modern shape variables $s_{ij}=u_{ij}^{2a}-1$ (with $u_{ij}=r_{ij}/r_0$ distances normalized by a scaling $r_0$ ) imply the quadratic relations $s_{ij}s_{kl}-s_{ik}s_{jl}=0$ together with a kernel condition $m_k s_{ik}\delta_l = m_l s_{il}\delta_k$ for a generator $\Delta=(\delta_i)$ of the kernel of the configuration matrix. The quadratic relations are exactly the defining equations of the degree-2 Veronese variety of algebraic geometry — the image of projective space under the map sending a point to all its degree-2 monomials, cut out by the 2x2 minors of a symmetric matrix of indeterminates.

Moeckel (2001) used this determinantal structure to prove generic finiteness of Dziobek configurations for the Newtonian potential; Dias (2017) extended generic finiteness to semi-integer exponents. The paper of Dias (arXiv:2601.07962, January 2026) identifies the Dziobek conditions with the Veronese variety, builds from them a quasi-projective 'Dziobek-Veronese variety' containing all Dziobek configurations, and applies the dimension-of-fibers theorem to the projection onto mass space, proving generic finiteness of Dziobek configurations for every homogeneous potential $U_a$ at once (the theorem excludes the exponent $a=0$ , where the shape map degenerates), and deriving a uniform upper bound $2^{\binom{n+1}{2}+n-1}$ for the number of classes for generic masses, independent of $a$ : a Bezout-type count (in the sense of the classical product-of-degrees bound on isolated solutions of a polynomial system) coming from the intersection of quadrics (via a Milnor-Thom-type bound on irreducible components), giving 8192 for n=4 versus the earlier estimate of 8472 attributed to Moeckel and Hampton. What the theorem does not provide is any information about the true size of these numbers: the bound is a degree count, expected to be far from sharp, and no exact count of Dziobek central configurations for generic (non-symmetric) masses is known for any $n\geq 4$ . Exact enumerations exist only inside imposed symmetry: e.g. bifurcation studies for equal-mass four-body families (Rusu and Santoprete, 2016), and the author's own follow-up with Tsai (Dias and Tsai, 2026), which for a symmetric family with $d$ unit masses on a regular simplex plus two axial masses and the logarithmic potential determines, via Gröbner bases, cylindrical algebraic decomposition, and Hermite root counting, that generically there are between 1 and 5 such configurations depending on the connected component of the generic mass set — showing both that exact counts are computable in principle and that they are mass-cell dependent.

Problem Statement

For the homogeneous potential $U_a$ (power-law case $a\in\mathbb{R}\setminus\{0,-1\}$ , including the Newtonian exponent $a=-3/2$ , together with the logarithmic case $a=-1$ ), and for each $n\geq 4$ , determine the sharp quantitative count of equivalence classes of Dziobek central configurations (configurations of the $n$ bodies whose affine hull has dimension exactly $n-2$ ) for generic positive masses, i.e. for mass vectors outside the exceptional algebraic subvariety supplied by the generic-finiteness theorem of the source paper. This task asks for the $n=4$ case of that family-wide count together with the two structural questions below, each of the three parts being of independent standing and separately answerable. Concretely: (1) determine the exact number of classes on each connected component of the generic mass set for $n=4$ (the planar four-body problem) — where the current bounds are the uniform $2^{13}=8192$ of the source paper and the earlier 8472 estimate, and where the true generic count is expected to be orders of magnitude smaller — while the analogous exact count for general $n\geq 4$ is the eventual goal of the program rather than part of this question; (2a) decide whether the uniform bound $2^{\binom{n+1}{2}+n-1}$ can be replaced by a sharp bound of strictly smaller order — a question subsumed by (1) only once (1) is settled for all $n$ and hence not made redundant by the $n=4$ case alone — and (2b) whether the sharp generic count is independent of the exponent $a$ , as the uniformity of the existing bound suggests. Answers must count classes modulo translation, rotation, and dilation, and must handle all degeneracy strata of the Dziobek equations within the count, including the strata of positive degeneracy index in the sense of the source paper (the degeneracy index counts the vanishing diagonal shape variables $s_{jj}$ and is always at most $n-2$ ).

References

Thiago Dias, 'The Veronese Geometry of Dziobek Configurations and Generic Finiteness for Homogeneous Potentials', arXiv:2601.07962, January 2026, DOI 10.48550/arXiv.2601.07962, https://arxiv.org/abs/2601.07962
Richard Moeckel, 'Generic finiteness for Dziobek configurations', Transactions of the American Mathematical Society 353 (2001) 4673-4686, DOI 10.1090/S0002-9947-01-02828-8, https://doi.org/10.1090/S0002-9947-01-02828-8
Marshall Hampton and Richard Moeckel, 'Finiteness of relative equilibria of the four-body problem', Inventiones mathematicae 163 (2006) 289-312, DOI 10.1007/s00222-005-0461-0, https://doi.org/10.1007/s00222-005-0461-0
Alain Albouy and Vadim Kaloshin, 'Finiteness of central configurations of five bodies in the plane', Annals of Mathematics 176 (2012) 535-588, DOI 10.4007/annals.2012.176.1.10, https://doi.org/10.4007/annals.2012.176.1.10
Marshall Hampton and Anders Jensen, 'Finiteness of spatial central configurations in the five-body problem', Celestial Mechanics and Dynamical Astronomy 109 (2011) 321-332, DOI 10.1007/s10569-010-9328-9, https://doi.org/10.1007/s10569-010-9328-9
Thiago Dias, 'New equations for central configurations and generic finiteness', Proceedings of the American Mathematical Society 145 (2017) 3069-3084, DOI 10.1090/proc/13427, https://doi.org/10.1090/proc/13427
Thiago Dias and Ya-Lun Tsai, 'Generalized Laura-Andoyer equations and the enumeration of some symmetrical classes of Dziobek configurations', arXiv:2606.31582, June 2026, DOI 10.48550/arXiv.2606.31582, https://arxiv.org/abs/2606.31582
Steve Smale, 'Mathematical problems for the next century', The Mathematical Intelligencer 20 (1998) 7-15, DOI 10.1007/bf03025291, https://doi.org/10.1007/bf03025291
David Rusu and Manuele Santoprete, 'Bifurcations of Central Configurations in the Four-Body Problem with Some Equal Masses', SIAM Journal on Applied Dynamical Systems 15(1) (2016) 440-458, DOI 10.1137/151002484, https://doi.org/10.1137/151002484