New equations for central configurations and generic finiteness

TL;DR

This paper establishes conditions under which the number of central configurations in the n-body problem is finite for generic masses, introduces new polynomial equations for these configurations, and explores their algebraic properties.

Contribution

It proves finiteness results for central configurations outside a Zariski closed set and derives new polynomial equations characterizing these configurations.

Findings

Finite number of central configurations for generic masses when n≥4.

Derived trilinear polynomial equations of degree 3 for fixed-dimension configurations.

Showed that mutual distances of k-dimensional configurations lie in a determinantal algebraic set.

Abstract

We consider the finiteness problem for central configurations of the $n-$body problem. We prove that, for $n\geq4$, there exists a (Zariski) closed subset $B$ in the mass space $\mathbb{R}^{n}$, such that if $(m_1,...,m_n) \in \mathbb{R}^n\setminus B$, then there is a finite number of corresponding classes of $(n-2)-$dimensional central configurations for potential associated to a semi-integer exponent. Also, we obtain trilinear homogeneous polynomial equations of degree $3$ for central configurations of fixed dimension and, for each integer $k \geq 1$, we show that the set of mutual distances associated to a $k-$dimensional central configuration is contained in a determinantal algebraic set.