The Veronese Geometry of Dziobek Configurations and Generic Finiteness for Homogeneous Potentials

TL;DR

This paper proves the generic finiteness of Dziobek central configurations for homogeneous potentials, derives a uniform upper bound on their number based on geometric properties, and compares it favorably to previous estimates.

Contribution

It introduces the Dziobek-Veronese variety, applies algebraic geometry techniques to analyze configuration counts, and provides a potential-independent bound for Dziobek configurations.

Findings

Fibers of the configuration-to-mass projection are finite outside a proper algebraic subvariety.

Derived a Bezout-type upper bound for the number of configurations, depending only on dimension.

For four bodies, the bound is 8192, lower than previous estimates.

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.