Brehm-Wintner-Conley Dimension, Pl\"ucker Coordinates, and Generalized Dziobek-Williams Equations for Central Configurations

Abstract

We develop an algebraic framework for central configurations of the $n$-body problem with homogeneous potentials, grounded in the exterior algebra of the configuration space and the normalized shifted Brehm--Wintner--Conley (BWC) matrix $S$. Relating the kernel of $S$ to the Pl\"ucker coordinates of the configuration, we generalize the determinantal equations obtained by Williams (1938) for the planar five-body problem to central configurations of any dimension and any number of bodies, and derive the Dziobek--Williams equations $\det(S_I^J)=\kappa\, z_I z_J$, which exhibit the compound matrix $S^{(t)}$ as a rank-one matrix. Introducing the \emph{Brehm--Wintner--Conley dimension} $\operatorname{bwc}(x)=\operatorname{rank}(S)$ (an integer invariant that stratifies central configurations and measures vertical degeneracy), together with the Pl\"ucker--BWC coordinates attached to it, we prove that each stratum of central configurations with fixed dimension and Brehm--Wintner--Conley dimension admits a base-point-free map into a Veronese variety, factoring through a Grassmannian invariant; on the Dziobek stratum this recovers the Dziobek--Veronese geometry previously introduced by the author. We further describe universal determinantal relations satisfied by the minors of $S$ and expand explicitly the resulting systems for the planar five- and six-body problems. We also interpret the mass-weighted entries of $S$ as an equilibrium stress: under the MacMillan--Bartky sign condition a strictly convex central configuration underlies a cable--strut tensegrity, and a theorem of Connelly then forces $S$ to be negative semidefinite with nullity three, so that vertical degeneracy cannot occur in this regime.