The Veronese Geometry of Dziobek Configurations and Generic Finiteness for Homogeneous Potentials
Thiago Dias
TL;DR
The paper establishes generic finiteness for Dziobek central configurations in the $n$-body problem with a homogeneous potential $U_a$ by linking Dziobek conditions to a Veronese-determinantal variety and applying the dimension-of-fibers theorem to mass-projection maps. A uniform Bezout-type bound $2^{\binom{n+1}{2}+n-1}$ on the number of such configurations is derived, independent of $a$, with a concrete improvement in the planar four-body case to $8192$. The approach highlights that the ambient configuration-space geometry governs the complexity of counting configurations more than the specific potential nonlinearity. The framework unifies and extends known finiteness results by exploiting Veronese geometry and fiber-dimension theory. Overall, the work provides a robust, geometry-driven method to quantify central configurations generically across homogeneous potentials.
Abstract
The main contribution of this paper is the proof of the generic finiteness of Dziobek central configurations for a homogeneous potential and the derivation of a uniform upper bound for their number. By exploiting the isomorphism between the Veronese variety and the determinantal variety associated with the Dziobek conditions, we define the Dziobek-Veronese variety and apply the dimension of fibers theorem to analyze the projection from the space of configurations and masses to the space of masses. We prove that the fibers of this projection, representing the central configurations for a given mass vector, are finite for masses chosen outside a proper algebraic subvariety. Furthermore, we utilize that the Dziobek variety is defined by an intersection of quadrics to obtain a bound of Bezout type for the number of Dziobek configurations with fixed masses given by a power of $2$ with exponent quadratic in $n$. Unlike previous estimates tailored to specific potentials, this bound depends solely on the dimension $n$. For instance, for the four-body problem, our bound reduces to $8192$, which is lower than the bound of $8472$ established by Moeckel and Hampton. This suggests that the complexity of counting Dziobek configurations for generic masses is governed primarily by the ambient geometry of the configuration space, rather than by the non-linearity of the interaction potential.