A 24 point Atiyah singular configuration in hyperbolic 3-space
Year: 2026
Type: preprint
Abstract: We construct a configuration of 24 distinct, noncoplanar points in hyperbolic 3-space whose Atiyah polynomial coefficient matrix has rank exactly 23. In the curvature -1 Beltrami-Klein ball, the points are q_i = (s_*/10000)v_i, with explicitly listed integer vectors v_i. A rational interval of width 10^{-30} isolates a unique zero s_* of a real analytic determinant. Reflection symmetry reduces the complex coefficient matrix to a real matrix. Outward-rounded interval arithmetic certifies opposite endpoint signs and a strictly negative derivative throughout the interval, proving existence, uniqueness, and simplicity of the zero. A second implementation, using a different arithmetic backend, polynomial basis, and pivot strategy, reproduces these bounds. The construction gives the upper bound 24 for the least cardinality of a singular hyperbolic Atiyah configuration.
Authors: Lyu, Yingjie, Ma, Jiming, Tan, Hu, Zhang, Ying
Institutions: Soochow University, Fudan University, Chinese Academy of Sciences
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221795449
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