Linear Invariants of Homogeneous Quasi-Translations in Dimension Five
Year: 2026
Type: preprint
Abstract: A polynomial map x+H:C^5->C^5 is a quasi-translation when its inverse is x-H. A question explicitly recorded as unsolved by de Bondt in 2018 asks whether every homogeneous quasi-translation in dimension five has a nonconstant linear invariant, equivalently whether the components H_1,...,H_5 are linearly dependent over C. This preprint proves an affirmative answer using de Bondt's published dimension-five structural reduction. The proof associates to the reduced map a projective curve C in P^3, constructs a relative secant divisor in P^1 x P^3, derives a directional differential identity for its bihomogeneous defining equation, and converts that identity into a polynomial invariant. A boundary-versus-genuine-secant trichotomy for an irreducible factor of this invariant contradicts the assumed absence of linear invariants. No degree bound is imposed. The accompanying supplement provides the manuscript source, a six-object proof map, a published-dependency ledger, hostile-review notes, licenses, an integrity/structure regression script, and SHA-256 hashes. The script is not a formal proof certificate. Independent specialist review and priority certification remain separate questions.
Author: Atwell, Casey
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212021446
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