Existence of continua of $(n-2)$-dimensional central configurations with mixed-sign masses for homogeneous potentials

This problem concerns the open ends left by Dias (Proc. Amer. Math. Soc. 145, 2017) in the study of Dziobek configurations, i.e. $(n-2)$-dimensional central configurations of the $n$-body problem with homogeneous potentials $U_a$. Of the remarks recorded in that paper, the resultant/cyclotomic obstruction concerns only an alternative proof technique (the same paper's Theorem 6.8 already proves generic finiteness for every semi-integer exponent, and Dias's 2026 Veronese-geometry preprint states generic finiteness for all real exponents $a\neq 0$ over the real mass space, outside an exceptional algebraic locus), and the 'extraneous components' remark is auxiliary. The open core is the paper's Remark 6.9: do continua of $(n-2)$-dimensional central configurations with mixed-sign mass vectors exist for the potentials $U_a$, and can they be produced through points of $W_3^a\cap\widetilde V^a$ of local dimension greater than $n$? Adjacent results (Roberts' planar five-body continuum with one negative mass; Alfaro-Perez-Chavela's charged four-body continua with positive masses and charges of both signs) do not settle the Dziobek case. The question remains open.

Background

For $n$ point particles at positions $x_1,\dots,x_n$ in $\mathbb{R}^d$ with masses $m_1,\dots,m_n$, consider the family of homogeneous pair potentials

$U_a(x)=\sum_{i<j}\frac{m_i m_j}{2a+2}\,r_{ij}^{2a+2},\qquad 2a\in\mathbb{Z},\; a\neq-1,$

where $r_{ij}=\lVert x_i-x_j\rVert$ and $a=-3/2$ gives the Newtonian $n$-body problem (the logarithmic potential replaces this at $a=-1$). A configuration $x$ is a central configuration if there is a constant $\lambda\neq 0$ with $\sum_{i\neq j}m_i r_{ij}^{2a}(x_i-x_j)+\lambda(x_j-c)=0$ for every $j$, where $c$ is the center of mass: the gravitational force on each particle is then proportional to its displacement from $c$, so such configurations are exactly the initial shapes of homographic (e.g. rigidly rotating) solutions. Central configurations are counted up to translation, rotation, reflection, and dilation; whether the number of classes is finite for every choice of positive masses is the classical finiteness problem, which appears on Smale's 1998 list of problems for the twenty-first century. It is fully settled only for $n=3$, with finiteness for all positive masses known for planar four-body configurations (Hampton-Moeckel). A configuration is named by its dimension $\delta(x)$, the dimension of the smallest affine space containing the particles; the $(n-2)$-dimensional central configurations are called Dziobek configurations (for $n=4$ these are exactly the planar ones). Moeckel proved that for $n\geq 4$ the Newtonian Dziobek configurations are generically finite: there is a proper Zariski closed subset $B$ of the positive mass cone such that every mass vector outside $B$ admits only finitely many classes. Dias extended this to every semi-integer exponent $a\neq 0$ as follows. Introducing the coordinates $(r,z,\Delta_0,m)$ — the mutual distances, auxiliary quantities $z_i$ arising from the Dziobek equations, a scaling variable $\Delta_0$, and the masses — one defines a quasi-affine algebraic set $\widetilde V^a\subset\mathbb{C}^{q+2n+1}$ ($q=n(n-1)/2$) by polynomial equations $g_{ij}^a=0$ expressing the Dziobek relations between the mutual distances and the $z_i$, the Cayley-Menger determinant $F=0$ (the determinant of the bordered $(n+1)\times(n+1)$ matrix of squared mutual distances, whose vanishing encodes affine dimension $n-2$), and moment equations $\Gamma_l=0$ expressing the central-configuration equations in mutual distances; every Dziobek central configuration maps into it. Degenerate loci $W_1$ (some $r_{ij}=0$), $W_2$ (vanishing Cayley-Menger cofactors $F_{ij}=0$), and $W_3^a$ (simultaneous vanishing of the polynomials $\Psi_i^a$, linear forms in the $z_i$ whose coefficients are products of mutual distances times partial derivatives of $F$) are removed to obtain $V^a$, and a Jacobian-criterion computation gives $\dim V^a\leq n$, from which generic finiteness of the fibres of the projection to mass space follows (Theorem 6.8). The positivity of the masses enters this argument essentially: it is what forces the $\Psi_i^a$ not to vanish simultaneously at configuration points (Propositions 4.3 and 4.4, whose proofs reduce to strict positivity of masses, distances, and a nonzero constant $\alpha$). In Remark 6.9 the author observes that with at least one negative mass these propositions fail, so no dimension bound on $\widetilde V^a$ near $W_3^a$ is available, and conjectures that continua of Dziobek configurations with mixed-sign masses might be produced by searching points $P\in W_3^a\cap\widetilde V^a$ with $\dim_P\widetilde V^a>n$, whose fibres over the mass space would then be positive-dimensional. A subsequent preprint of Dias (the Veronese-geometry paper, arXiv:2601.07962) reframes the construction through the Veronese embedding — the map sending a vector of mutual distances to all its monomials of a fixed degree, which turns the algebraic analysis of $\widetilde V^a$ into the study of a Dziobek-Veronese variety — and states generic finiteness with a uniform Bezout-type bound $2^{\binom{n+1}{2}+n-1}$ for all real exponents $a\neq 0$ and all real mass vectors outside an exceptional algebraic locus; since this is exactly a generic statement over the real mass space, any mixed-sign continuum would have to live on the exceptional locus, and the existence question is untouched. Nearby degeneracy phenomena are known: Roberts constructed a continuum of planar relative equilibria for the five-body Newtonian problem with one negative mass (persisting for the logarithmic point-vortex potential), Alfaro and Perez-Chavela found continua of central configurations in charged four-body problems with positive masses carrying electric charges of both signs (so that the effective pair interactions are of mixed sign, but not signed masses in a potential $U_a$), and Santoprete exhibited a three-body continuum for the harmonic potential $U=MI/2$, which in the normalization above is the degenerate exponent $a=0$ at which every configuration is central. None of these produces a continuum in the Dziobek class for a potential $U_a$ with $a\neq 0$ and genuinely signed masses.

Problem Statement

For $n\geq 4$ and a semi-integer exponent $a$ with $2a\in\mathbb{Z}$ and $a\neq 0$ (at $a=-1$, $U_a$ denotes the logarithmic potential, as in the Background), determine whether there exists a mixed-sign mass vector $m=(m_1,\dots,m_n)\in\mathbb{R}^n$ — at least one positive and at least one negative component, total mass $M\neq 0$, all mutual distances nonzero — for which the set of equivalence classes of $(n-2)$-dimensional central configurations of the potential $U_a$ contains a continuum. The two alternatives are a continuum of equivalence classes for a single exponent $a$ and a single mixed-sign mass vector $m$, and finiteness of the set of equivalence classes for every semi-integer exponent $a\neq 0$, every $n\geq 4$, and every mixed-sign mass vector $m\in\mathbb{R}^n$; the exceptional locus of the mass space is not excluded, since the finiteness claimed for all real exponents $a\neq 0$ outside a proper algebraic locus of the real mass space by the Veronese-geometry preprint of Dias leaves that locus open. Remark 6.9 of Dias 2017 also asks, in the same mixed-sign setting and as a question in its own right, whether continua of $(n-2)$-dimensional central configurations with mixed-sign masses can be constructed through points $P\in W_3^a\cap\widetilde V^a$ with $\dim_P\widetilde V^a>n$ whose fibres under the projection $(r,z,\Delta_0,m)\mapsto m$ are positive-dimensional, where $\widetilde V^a$ and $W_3^a$ are the loci defined in the Background. The first open case is $n=4$, where $(n-2)$-dimensional means planar.

References

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