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New equations for central configurations and generic finiteness


Author: Thiago Dias
Journal: Proc. Amer. Math. Soc. 145 (2017), 3069-3084
MSC (2010): Primary 70F10, 70F15, 37N05, 14A10
DOI: https://doi.org/10.1090/proc/13427
Published electronically: January 6, 2017
MathSciNet review: 3637954
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Abstract: We consider the finiteness problem for central configurations of the $n$-body problem. We prove that, for $n\geq 4$, there exists a (Zariski) closed subset $B$ in the mass space $\mathbb {R}^{n}$, such that if $(m_1,\dots ,m_n) \in \mathbb {R}^n\setminus B$, 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.


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Thiago Dias
Affiliation: Departamento de Matemática, Universidade Federal Rural de Pernambuco - Rua Dom Manuel de Medeiros s/n, 52171-900, Recife, Pernambuco, Brasil
Email: thiago.diasoliveira@ufrpe.br

Keywords: $n$-body problem, central configuration, celestial mechanics, Jacobian criterion, Cayley-Menger matrix
Received by editor(s): January 22, 2016
Received by editor(s) in revised form: June 18, 2016, and August 8, 2016
Published electronically: January 6, 2017
Communicated by: Yingfei Yi
Article copyright: © Copyright 2017 American Mathematical Society