Abstract
We strengthen a generic finiteness result due to Moeckel by showing that the number of spatial central configurations of the Newtonian five-body problem with positive masses is finite, apart from some explicitly given special cases of mass values.
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Albouy A., Chenciner A.: Le problème des n corps et les distances mutuelles. Inv. Math. 131, 151–184 (1998)
Albouy, A., Kaloshin, V.: Personal communication (2010)
Bernstein D.N.: The number of roots of a system of equations. Funct. Anal. Appl. 9, 183–185 (1975)
Bieri R., Groves J.R.J.: The geometry of the set of characters induced by valuations. J. Reine Angew. Math. 347, 168–195 (1984)
Decker, W., Greuel, G.M., Pfister, G., Schönemann, H.: Singular 3-1-1—A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2010)
Dziobek O.: Über einen merkwürdigen fall des vielkörperproblems. Astron. Nach. 152, 152 (1900)
Euler L.: De motu rectilineo trium corporum se mutuo attrahentium. Novi. Comm. Acad. Sci. Imp. Petrop. 11, 144–151 (1767)
Hampton M., Moeckel R.: Finiteness of relative equilibria of the four-body problem. Inv. Math. 163, 289–312 (2006)
Hept K., Theobald T.: Tropical bases by regular projections. Proc. Am. Math. Soc. 137, 2233–2241 (2009)
Jensen, A.N.: Algorithmic Aspects of Gröbner Fans and Tropical Varieties. PhD thesis, Department of Mathematical Sciences, University of Aarhus, Denmark (2007)
Jensen, A.N.: Gfan, a software system for Gröbner fans. http://www.math.tu-berlin.de/~jensen/software/gfan/gfan.html (2009)
Khovanskii A.G.: Newton polyhedra and toric varieties. Funct. Anal. Appl. 11, 289–296 (1977)
Kushnirenko A.G.: Newton polytopes and the bézout theorem. Funct. Anal. Appl. 10, 233–235 (1976)
Lagrange, J.L.: Essai sur le problème des trois corps. Oeuvres 6 (1772)
Lee T.L., Santoprete M.: Central configurations of the five-body problem with equal masses. Celest. Mech. Dyn. Astron. 104, 369–381 (2009)
Moeckel R.: On central configurations. Math. Zeit. 205, 499–517 (1990)
Moeckel R.: Generic finiteness for Dziobek configurations. Trans. Am. Math. Soc. 353, 4673–4686 (2001)
Moulton F.R.: The straight line solutions of the problem of n bodies. Ann. Math. 12, 1–17 (1910)
Newton I.: Philosophi Naturalis Principia Mathematica. Royal Society, London, UK (1687)
Prajna, S., Papachristodoulou, A., Seiler, P., Parrilo, P.A.: SOSTOOLS: Sum of squares optimization toolbox for MATLAB (2004)
Roberts G.: A continuum of relative equilibria in the five-body problem. Phys. D 127, 141–145 (1999)
Roberts, G.: Personal communication (2009)
Speyer D., Sturmfels B.: The tropical grassmannian. Adv. Geom. 4, 389–411 (2004)
Stein, W., et al.: Sage Mathematics Software (Version 4.5.2). The Sage Development Team http://www.sagemath.org (2010)
Stengle G.: A nullstellensatz and a positivstellensatz in semialgebraic geometry. Math. Ann. 207, 87–97 (1974)
Sturmfels B.: Gröbner Bases and Convex Polytopes. American Mathematical Society, Providence, R.I. (1996)
Sturmfels, B.: Solving systems of polynomial equations, CBMS Regional Conference Series in Mathematics, vol 97. Published for the Conference Board of the Mathematical Sciences, Washington, DC (2002)
Witty, C.: Isolate Real Roots of Real Polynomials, http://www.sagemath.org/doc/reference/sage/rings/polynomial/real_roots.html (2009)
Zeigler G.: Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)
Acknowledgments
Some of our computations were run on one of the Sage Foundation’s 24-core Sun X4450s, supported by National Science Foundation Grant No. DMS-0821725. We were also both supported by the American Institute of Mathematics. Furthermore, the second author was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Göttingen.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Hampton, M., Jensen, A. Finiteness of spatial central configurations in the five-body problem. Celest Mech Dyn Astr 109, 321–332 (2011). https://doi.org/10.1007/s10569-010-9328-9
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DOI: https://doi.org/10.1007/s10569-010-9328-9