Finiteness of planar central configurations of five bodies for all positive masses: On Smale's sixth problem
Abstract
Combining local boundary analysis, algebraic curves and the computational statements specified in the appendix, we prove that the planar Newtonian five-body problem has finitely many central configurations up to rotation, translation and scaling for every choice of positive masses. This gives the five-body statement of Smale's sixth problem. We use the singular-sequence classification of Albouy and Kaloshin and resolve the corresponding boundary equations through weighted blow-up charts, rank calculations and local compatibility conditions. For the two remaining diagrams, we prove that the product Δ of all squared mutual distances vanishes. On a hypothetical solution curve containing a positive arc, a pole of a four-product therefore gives a zero of Δ. A pole of Δ elsewhere on the completed curve, together with Δ4=∏i<jpij, then contradicts the same local vanishing result. The computational statements for Diagrams~2, 3 and~6 involve exact algebra, interval arithmetic and ball arithmetic.
Related
- Finiteness of planar central configurations of five bodies with positive masses
- Finiteness of planar central configurations of six bodies for all positive masses: Part I, exceptional-mass first limits
- Finiteness of Dziobek central configurations in the Newtonian five-body problem
- Uniqueness of convex central configurations in the planar (1 + 4)-body problem
- Planar Central Configurations of Four Equal Masses: Classification and Morse Indices
BibTeX
@misc{chen2026finiteness,
title = {{Finiteness of planar central configurations of five bodies for all positive masses: On Smale's sixth problem}},
author = {Zhe Chen and Zhicheng Tong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16107},
eprint = {2610.16107},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16107},
}
RIS
TY - UNPB
TI - Finiteness of planar central configurations of five bodies for all positive masses: On Smale's sixth problem
AU - Zhe Chen
AU - Zhicheng Tong
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.16107
UR - https://arxiv.org/abs/2610.16107
AB - Combining local boundary analysis, algebraic curves and the computational statements specified in the appendix, we prove that the planar Newtonian five-body problem has finitely many central configurations up to rotation, translation and scaling for every choice of positive masses. This gives the five-body statement of Smale's sixth problem. We use the singular-sequence classification of Albouy and Kaloshin and resolve the corresponding boundary equations through weighted blow-up charts, rank calculations and local compatibility conditions. For the two remaining diagrams, we prove that the product Δ of all squared mutual distances vanishes. On a hypothetical solution curve containing a positive arc, a pole of a four-product therefore gives a zero of Δ. A pole of Δ elsewhere on the completed curve, together with Δ^4=∏_i<jp_ij, then contradicts the same local vanishing result. The computational statements for Diagrams~2, 3 and~6 involve exact algebra, interval arithmetic and ball arithmetic.
ER -