Finiteness of Planar Central Configurations for Five Positive Masses
Source: hexagonmath.org/abs/2610.00467v1
We prove that, for any five positive masses, there are finitely many planar Newtonian central configurations up to translations, rotations, and positive scaling. Building on the singular-diagram framework of Albouy and Kaloshin, we treat the exceptional masses left open by their finiteness theorem. The proof retains a lower bound for the physical potential after passing to the complex algebraic equations. On each fixed potential level, energy estimates, torque identities, and local response calculations control the possible degenerations. A polynomial selection argument then yields…
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