Sharp Stability and Geometric Classification of a Vámos Quartic Deformation
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Abstract
We study the quartics h_λ=4e_1e_3+λ a^2b^2, 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 A_1 and one of type A_3 in the interior; one additional A_1 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.
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Research details
- Subjects
- Mathematics · Geometry and Optimal Transport
- Keywords
- Vámos quartic, real-stable polynomials, hyperbolicity cones, projective classification, singularities, Lorentz facets
- Manuscript date
- 2026-10-22
- Reader reuse
- No additional permission granted
Scope
Exact stability interval [0,64/27] and positive-parameter projective, cone-symmetry and dual-facet classification for the specified Vámos quartic coefficient deformation, including the endpoint transition.
Disclosures
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.
Reproduction
The paper has 11 pages. The editable-source archive contains 6 files. Compile main.tex. The standard-library verify.py checks the exact polynomial and incidence identities accompanying the analytic classification.