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The number of planar central configurations of four bodies

A mix of human-written and AI-generated textHuman understanding: all partsmath.DS — Dynamical Systemsmath-ph — Mathematical Physicsmath.NA — Numerical Analysis

Contributed by Tejasvi Singh Tomar ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

For four positive masses in the plane we determine the number NN of central configurations, counted up to rotation, translation and scaling, in terms of one fold surface, whose certified enclosures decide NN for all masses outside a closed null set. Of these configurations, 1212 are collinear, by Moulton's theorem, and 66 are convex, by the existence theorem of MacMillan and Bartky and the uniqueness proved in the author's preprint arXiv:2609.35632. Through Dziobek's relations, the concave configurations with inner body ii form the fibre of a proper mass map of degree 22. Their number cic_i, counted up to reflection, is 11, 22, 33 or 44, and we read it off from the position of the mass ratios relative to this surface. Altogether N=18+2(c1+c2+c3+c4)N=18+2(c_1+c_2+c_3+c_4), an even number with 32≤N≤5032\le N\le50. Moreover N=32N=32 exactly at Palmore's masses, and every even value in this range occurs. For four equal masses we recover Albouy's count N=50N=50. The proof shows with interval certificates that the mass map is injective where its Jacobian is not positive, obtains its degree from the restricted problem, and analyses its folds, cusps and Palmore's umbilic. A formalization in Lean also proves these results.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23256909

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Version history

  1. v1 · 79 pagesSubmitted by Tejasvi Singh TomarPublished on Zenodo: https://doi.org/10.5281/zenodo.23256909CurrentNov 12, 2026