Toward finiteness of central configurations for the planar six-body problem by symbolic computations

In this paper we develop symbolic computation algorithms to investigate finiteness of central configurations for the planar $n$-body problem. Our approach is based on Albouy-Kaloshin's work on finiteness of central configurations for the 5-body problems. In their paper, bicolored graphs called $zw$-diagrams were introduced for possible scenarios when the finiteness conjecture fails, and proving finiteness amounts to exclusions of central configurations associated to these diagrams. Following their method, the amount of computations becomes enormous when there are more than five bodies. Here we introduce matrix algebra for determination of both diagrams and asymptotic orders, devise several criteria to reduce computational complexity, and verify finiteness mostly through automated deductions. For the planar six-body problem, our first algorithm effectively narrows the proof for finiteness down to 117 $zw$-diagrams, the second algorithm eliminates 31 of them, the last algorithm eliminates 62 other diagrams except for masses in some co-dimension 2 variety in the mass space, and leaving 24 cases unsolved.