Finiteness of relative equilibria of the four-body problem

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. Supplementary material is available in the online version of this article at and is accessible for authorized users. Mathematics Subject Classification (2000) 70F10, 70F15, 37N05, 76Bxx