Complex solution sets of the planar five-body problem at special masses

Abstract

Albouy and Kaloshin observed that the polynomial system for planar central configurations of five bodies has a continuum of solutions at the masses (4,4,4,4,1), obtained from Roberts' continuum at (4,4,4,4,−1). At these masses, finiteness of the central configurations cannot follow from the complexified equations alone. We study the complex solution sets at three special mass vectors. At (1,4,4,4,4) the solution set modulo rotations is the union of 36 explicit, pairwise disjoint irreducible curves, 18 of genus 3 and 18 rational, and a finite set. The curves contain the modified Roberts continuum, but none of their points is a positive central configuration. At (1,1,4,4,4) the solution set modulo rotations is finite. At (9,36,36,36,4) the potential vanishes identically on every curve in the solution set. These masses, their positive multiples and their permutations therefore have finitely many central configurations, and we prove this without counting them. Along a curve of solutions the valuations of the mutual distances at the places at infinity sum to zero. Following Albouy and Kaloshin, we exclude such curves with linear functionals on the possible valuation vectors. The new inputs concern their limit diagrams, numbered as in Fig. 11 of their paper. An expansion at places of diagram 3 shows that a curve with such a place is one of the 36 curves. A second expansion excludes the places of diagram 6 that the mass conditions allow on the rays through (1,1,4,4,4), and case analyses treat diagram 5. We also show that at positive masses, places of diagrams 1 and 4 occur only for masses on finitely many explicit curves. Apart from a few exact polynomial computations, which are in the supplementary material, our proofs are by hand, although some of the results that we cite rest on computations.