Toward finiteness of central configurations for the planar six-body problem by symbolic computations. (I) Determine diagrams and orders

In a series of papers 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. In this paper we introduce matrix algebra for determination of possible diagrams and asymptotic orders, devise several criteria to reduce computational complexity, and determine possible zw-diagrams by automated deductions. For the planar six -body problem, we show that there are at most 86 zw-diagrams.(c) 2023 Elsevier Ltd. All rights reserved.