Finding solutions with the Borsuk-Ulam theorem
Abstract
Many problems in nonlinear analysis, game theory, economics, and geometry have existence proofs based on Brouwer's fixed point theorem or the Borsuk--Ulam theorem. We study a common numerical approach: deform a problem with one known solution into the desired problem and follow its zeros. A constructive reduction of fixed points to odd spherical zeros makes the approach available to a broad class of problems, including finite-dimensional Galerkin systems, periodic solutions, Nash and economic equilibria, and geometric mass partitions, subject to the regularity and approximation requirements of each formulation. On the sphere, a linear projection supplies an explicit starting model with one antipodal pair of zeros. We give conditions for tracking the resulting curve, allowing turns in the deformation parameter, and relate computational work to accuracy and conditioning along the path. A family of linear models parametrized by antipodal pairs reaches every regular spherical solution from an open set of starts. This simple parameter space provides a basis for exploring multiple solutions; a finite search for all solutions needs additional coverage information. For mass partitions of finite point sets, each continuation step has cost proportional to the number of points in fixed dimension.
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BibTeX
@misc{musin2026finding,
title = {{Finding solutions with the Borsuk-Ulam theorem}},
author = {Oleg R. Musin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39092},
eprint = {2610.39092},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39092},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Finding solutions with the Borsuk-Ulam theorem
AU - Oleg R. Musin
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39092
UR - https://arxiv.org/abs/2610.39092
AB - Many problems in nonlinear analysis, game theory, economics, and geometry have existence proofs based on Brouwer's fixed point theorem or the Borsuk--Ulam theorem. We study a common numerical approach: deform a problem with one known solution into the desired problem and follow its zeros. A constructive reduction of fixed points to odd spherical zeros makes the approach available to a broad class of problems, including finite-dimensional Galerkin systems, periodic solutions, Nash and economic equilibria, and geometric mass partitions, subject to the regularity and approximation requirements of each formulation. On the sphere, a linear projection supplies an explicit starting model with one antipodal pair of zeros. We give conditions for tracking the resulting curve, allowing turns in the deformation parameter, and relate computational work to accuracy and conditioning along the path. A family of linear models parametrized by antipodal pairs reaches every regular spherical solution from an open set of starts. This simple parameter space provides a basis for exploring multiple solutions; a finite search for all solutions needs additional coverage information. For mass partitions of finite point sets, each continuation step has cost proportional to the number of points in fixed dimension.
ER -