Using Gröbner Bases and Invariant Theory to
check a conjecture in Celestial Mechanics

Ilias Kotsireas
Universite Pierre et Marie Curie, Paris VI
LIP6 BP 168
4, place Jussieu, 75252 Paris cedex 05
France
E-mail:Ilias.Kotsireas@lip6.fr

Abstract

The N-body problem is one of the most widely studied problems of Celestial mechanics. We are interested in a conceptually simple but important class of solutions, called central configurations. They are the only solutions of the N-body problem that can be computed analytically. Geometrically they represent solutions where the configuration of the N bodies remains similar to itself during motion.

Much work has been done on the problem of central configurations. Many different mathematical tools and methodologies have been used succesfully, to compute central configurations in various contexts. Me mention briefly Morse Theory, Homology Theory, Determinants, Numerical Analysis by computer, and Lie transforms.

It has been proved by A. Albouy [#!Alb95!#], that every planar central configuration of the newtonian 4-body problem with equal masses, has at least one symmetry axis. This fact has been verified using Symbolic Computation methods by A. Albouy [#!Alb96!#] and the author [#!Kot!#]. Moreover, this constructive proof revealed the fact that there is only one central configuration with exactly one symmetry axis.

In both the above approaches, the symmetry of the equations has been used in different ways.

By using the method presented in [#!KL!#] we obtain a formulation of the equations of the planar 4-body problem with equal masses, which is adequate, after an easy symetrisation of the equations, for solving the problem via an Invariant Theory polynomial system solving algorithm, proposed by A. Colin [#!Col!#]. Ultimately, this algorithm makes use of Gröbner Bases computations. However, this method has the advantage that it can be applied in a more general context, without significant modifications. In particular we use this method to check the validity of the Albouy-Simó conjecture for the 4-body problem with equal masses.

Moreover we obtain some interesting by-products, namely some new formulae for a particular action of the symmetric group S₂ on 4 variables, involving binomial coeffcients and Newton-Gregory interpolation polynomials. The discovery of these formulae has been done using Gröbner Bases computations in GB [#!Fau!#] through its AXIOM interface and following the ideas exposed in [#!BE!#]. The on-line Encyclopedia of Integer Sequences, maintaned by N. J. A. Sloane [#!Slo!#], has been of great help.

It is interesting to note that in all cases considered, we obtain a more compact form for the traditional Hironaka decomposition. Indeed, the expressions obtained are of lower degree and with smaller coefficients. Only the application of an easy recursive algorithm is needed, to obtain the actual Hironaka decomposition. These results suggest that perhaps the best way to compute the Hironaka decomposition of big invariants, is to first decompose them in small parts, find a formulae for each part and then combine the pieces together. We have thus computed for example the Hironaka decompositions of the invariants

B¹⁵⁶b⁷⁶+B⁷⁶b⁷⁵D⁸⁰d+D⁷⁶d⁷⁵B⁸⁰b+D¹⁵⁶d⁷⁶

and

B²⁴¹b¹⁵⁹+B¹⁵⁶b⁷⁶D⁸⁵d⁸³+B¹⁶¹b¹⁵⁸ D⁸⁰d+B⁷⁶b⁷⁵D¹⁶⁵d⁸⁴+D⁷⁶d⁷⁵B¹⁶⁵ b⁸⁴+D¹⁶¹d¹⁵⁸B⁸⁰b+D¹⁵⁶d⁷⁶B⁸⁵b⁸³ +D²⁴¹d¹⁵⁹

by implementing the recursive algorithm in MAPLE. These frontier Invariant Theory computations take some minutes using this approach, whereas they are already by far unfeasible with the general-purpose systems, such as MAGMA or INVAR. In a forthcoming paper we plan to present algorithms which apply in a more general Invariant-Theoretic context. This more systematic study is joint work in progress with A. Colin.

The final computations to check the validity of the Albouy-Simó conjecture up to a certain range, have been performed in FGB [#!Fau!#], on a Bi-Pentium machine under Linux.