Research Article · Preprint Mathematics
Sharp Stability and Geometric Classification of a Vámos Quartic Deformation
- Primary subject
- Mathematics
- Subject subfield
- Geometry and Optimal Transport
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 1 · Permanent link
Overview
Abstract
We study the quartics hλ = 4e₁e₃ + λa²b², where the elementary symmetric polynomials are taken in four variables. We prove that their real-stability interval is exactly [0, 64/27]. Throughout its positive part, the parameter is a genuine complex projective modulus, and the full linear automorphism group of the positive hyperbolicity cone consists of positive scalings and the independent interchanges of the two distinguished variable pairs. These cones are consequently pairwise linearly nonisomorphic and nonhomogeneous, including at the endpoint. The projective surfaces have eight singularities of type A₁ and one of type A₃ in the interior; one additional A₁ point appears at the endpoint. Correspondingly, the dual cone has exactly four Lorentz facets in the interior and exactly five at the endpoint. The arguments give an exact classification for this coefficient deformation of the specialized Vámos quartic, using classical hyperbolicity, singularity, and tangent-cone methods.
Keywords Vámos quartic · real-stable polynomials · hyperbolicity cones · projective classification · singularities · Lorentz facets
Scope of the work
Sharp real stability for hλ = 4e₁e₃ + λa²b² exactly when 0 ≤ λ ≤ 64/27. The positive parameter interval has pairwise distinct complex projective classes and linearly nonisomorphic hyperbolicity cones, with endpoint-complete symmetry and dual-facet classification. The reducible zero-parameter member is distinguished from the positive classification. Full four-variable inverse-half-power positivity remains open. Classical methods and prior constructions are credited; worldwide priority remains provisional. The cited Wagner–Wei corrigendum was not retrieved in full, while the h₁ baseline is explicit in the later inspected Kummer source.
Reproduction notes
The six-member package contains the eleven-page main.pdf, complete main.tex, verify.py, reader README, citation metadata and SHA256SUMS.txt. The 57 exact public checks passed in normal and optimized execution, including a coefficient-mutation negative control. An isolated rebuild from the actual ZIP matched every page of PDF text and pixels. Algebraic and analytic proofs establish the parameter-wide conclusions; finite diagnostics do not replace them. Private assessments, internal reviews, correspondence and ongoing inverse-half-power research are excluded.
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AI-assisted tools supported literature discovery, proof development, independent adversarial checks, exact symbolic and rational-arithmetic verification, and manuscript preparation. These internal checks are not external human refereeing or formal proof-assistant certification. The author retains responsibility for statements, proofs and citations. Originality is qualified by the inspected literature and the stated source-access limit.
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Cite this edition
Scott Schubr. (2026). Sharp Stability and Geometric Classification of a Vámos Quartic Deformation (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/sharp-stability-geometric-classification-vamos-quartic-deformation/v/1
BibTeX
@techreport{luminary_sharp_stability_geometric_classification_vamos_quartic_deformation_v1,
title = {Sharp Stability and Geometric Classification of a Vámos Quartic Deformation},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/sharp-stability-geometric-classification-vamos-quartic-deformation/v/1}
}Loading discussion…