# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Finiteness of relative equilibria of the four-body problem. (English) Zbl 1083.70012
Summary: We show that the number of relative equilibria of the Newtonian four-body problem is finite, up to symmetry. In fact, we show that this number is always between 32 and 8472. The proof is based on symbolic and exact integer computations which are carried out by computer.

##### MSC:
 70F10 $n$-body problems 70-08 Computational methods (mechanics of particles and systems) 68W30 Symbolic computation and algebraic computation
Macaulay2
##### References:
 [1] Albouy, A.: Integral Manifolds of the N-body problem. Invent. Math. 114, 463–488 (1993) [2] Albouy, A.: The symmetric central configurations of four equal masses. In: Hamiltonian Dynamics and Celestial Mechanics, Contemp. Math. 198 (1996) [3] Albouy, A., Chenciner, A.: Le problème des n corps et les distances mutuelles. Invent. Math. 131, 151–184 (1998) [4] Avis, D.: lrs – Version 4.1. http://cgm.cs.mcgill.ca/vis/C/lrs.html [5] Avis, D., Fukuda, K.: A Pivoting Algorithm for Convex Hulls and Vertex Enumeration of Arrangements and Polyhedra. Discrete Comput. Geom. 8, 295–313 (1992) [6] Bernstein, D.N.: The number of roots of a system of equations. Funct. Anal. Appl. 9, 183–185 (1975) [7] Cabral, H.: On the integral manifolds of the n-body problem. Invent. Math. 20, 59–72 (1973) [8] Chazy, J.: Sur certaines trajectoires du problème des n corps. Bull. Astron. 35, 321–389 (1918) [9] Christof, T., Loebel, A.: PORTA: POlyhedron Representation Transformation Algorithm, Version 1.3.2. http://www.iwr.uni-heidelberg.de/wr/comopt/soft/PORTA/readme.html [10] Cox, D., Little, J., O’Shea, D.: Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. New York: Springer 1997 [11] Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry. New York: Springer 1998 [12] Dziobek, O.: Über einen merkwürdigen Fall des Vielkörperproblems. Astron. Nachr. 152, 33–46 (1900) [13] Emiris, I.: Mixvol. http://www.inria.fr/saga/emiris [14] Emiris, I., Canny, J.: Efficient incremental algorithm for the sparse resultant and the mixed volume. J. Symb. Comput. 20, 117–149 (1995) [15] Euler, L.: De motu rectilineo trium corporum se mutuo attrahentium. Novi Comm. Acad. Sci. Imp. Petrop. 11, 144–151 (1767) [16] Hampton, M.: Concave Central Configurations in the Four Body Problem. Thesis, University of Washington 2002 [17] Grayson, D.R., Sullivan, M.E.: Macaulay 2, a software system for research in algebraic geometry and commutative algebra. http://www.math.uic.edu/Macaulay2/ [18] http://www.math.umn.edu/ick [19] Huber, B., Sturmfels, B.: A polyhedral method for solving sparse polynomial systems. Math. Comput. 64, 1541–1555 (1995) [20] Khovansky, A.G.: Newton polyhedra and toric varieties. Funct. Anal. Appl. 11, 289–296 (1977) [21] Kushnirenko, A.G.: Newton polytopes and the Bézout theorem. Funct. Anal. Appl. 10, 233–235 (1976) [22] Kuz’mina, R.P.: On an upper bound for the number of central configurations in the planar n-body problem. Sov. Math. Dokl. 18, 818–821 (1977) [23] Lagrange, J.L.: Essai sur le problème des trois corps. OEuvres, vol. 6 (1772) [24] Lefschetz, S.: Algebraic Geometry, Princeton: Princeton University Press 1953 [25] Lehmann-Filhés, R.: Über zwei Fälle des Vielkörpersproblems. Astron. Nachr. 127, 137–143 (1891) [26] Llibre, J.: On the number of central configurations in the N-body problem. Celest. Mech. Dyn. Astron. 50, 89–96 (1991) [27] MacDuffee, C.C.: Theory of Matrices. New York: Chelsea Publishing Co. 1946 [28] MacMillan, W.D., Bartky, W.: Permanent configurations in the problem of four bodies. Trans. Am. Math. Soc. 34, 838–875 (1932) [29] McCord, C.K.: Planar central configuration estimates in the n-body problem. Ergodic Theory Dyn. Syst. 16, 1059–1070 (1996) [30] McCord, C.K., Meyer, K.R., Wang, Q.: The integral manifolds of the three body problem. Providence, RI: Am. Math. Soc. 1998 [31] Moeckel, R.: Relative equilibria of the four-body problem. Ergodic Theory Dyn. Syst. 5, 417–435 (1985) [32] Moeckel, R.: On central configurations. Math. Z. 205, 499–517 (1990) [33] Moeckel, R.: Generic Finiteness for Dziobek Configurations. Trans. Am. Math. Soc. 353, 4673–4686 (2001) [34] Moeckel, R.: A Computer Assisted Proof of Saari’s Conjecture for the Planar Three-Body Problem. To appear in Trans. Am. Math. Soc. [35] Moulton, F.R.: The Straight Line Solutions of the Problem of n Bodies. Ann. Math. 12, 1–17 (1910) [36] Motzkin, T.S., Raiffa, H., Thompson, G.L., Thrall, R.M.: The double description method. Ann. Math. Stud. 28, 51–73 (1953) [37] Newton, I.: Philosophi naturalis principia mathematica. London: Royal Society 1687 [38] Roberts, G.: A continuum of relative equilibria in the five-body problem. Phys. D 127, 141–145 (1999) [39] Saari, D.: On the Role and Properties of n-body Central Configurations. Celest. Mech. 21, 9–20 (1980) [40] Shafarevich, I.R.: Basic Algebraic Geometry 1, Varieties in Projective Space. Berlin, Heidelberg, New York: Springer 1994 [41] Simó, C.: Relative equilibria in the four-body problem. Celest. Mech. 18, 165–184 (1978) [42] Smale, S.: Topology and Mechanics, II, The planar n-body problem. Invent. Math. 11, 45–64 (1970) [43] Smale, S.: Mathematical problems for the next century. Math. Intell. 20, 7–15 (1998) [44] Tien, F.: Recursion Formulas of Central Configurations. Thesis, University of Minnesota 1993 [45] Walker, R.: Algebraic Curves. New York: Dover Publications, Inc. 1962 [46] Wintner, A.: The Analytical Foundations of Celestial Mechanics. Princeton Math. Series 5. Princeton, NJ: Princeton University Press 1941 [47] Wolfram, S.: Mathematica, version 5.0.1.0. Wolfram Research, Inc. [48] Xia, Z.: Central configurations with many small masses. J. Differ. Equations 91, 168–179 (1991) [49] Xia, Z.: Central configurations for the four-body and five-body problems. Preprint [50] Zeigler, G.: Lectures on Polytopes. Grad. Texts Math. 152. New York: Springer 1995