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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
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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

DOI: https://doi.org/10.1090/proc/13427
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