Rotational Invariants Do Not Separate the Roots of a Quintic: Obstructions from Galois Symmetry, the Icosahedral Group, and Hypergeometric Monodromy
Year: 2026
Type: preprint
Abstract: We examine a proposed approach to the quintic in which the five roots are placed as points in three-dimensional space, a zeta function is built from the rotational invariants of this configuration, and the roots are to be recovered one at a time by analytic continuation of that zeta function. We show that this rotation–zeta separation method fails in general by exhibiting three mutually independent obstructions. First, any construction that does not label the roots is invariant under the Galois group; for z⁵ − z − 1, every rational expression in the symmetric invariants built from real coordinates lies in the fixed field of the centralizer D₄ of complex conjugation, which contains none of the four non-real roots (Theorem A). Second, rotational invariants determine the configuration of roots only up to rotation; consequently z⁵ − z − 1 and z⁵ − z + 1, which have no root in common, give rise to the same zeta function (Theorem B). Third, the roots of the Bring–Jerrard family x⁵ − x − t admit ₄F₃ hypergeometric representations, but the monodromy around t = ∞ is a 5-cycle; hence not even a single root can be selected continuously from data that are single-valued in t (Theorem C). We also verify that S₅ does not embed in O(3), and explain that the proper role of three-dimensional rotational symmetry for the quintic is the one identified by Klein's theory of the icosahedron. Supplementary files: verify_quintic.py and verify_proofs.py reproduce all numerical values and the computable proof steps (70 and 33 checks, all passing with Python 3.13.16, SymPy 1.14.0, mpmath 1.3.0); their outputs are included. The use of generative AI tools is disclosed in the manuscript.
Author: Ye, Hyunmin
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222866452
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