Sharp Stability and Geometric Classification of a Vámos Quartic Deformation
Loading original first page…
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.
Read the full paper in the original PDF.
Edition files
- main.pdf103.9 KiB · pdf
SHA-256
e5a7d8d6b50e2a2b28b7a6be2598ce4992ad11934a406ab463f7bc862493864c - Sharp_Stability_Publication.zip120.0 KiB · package
SHA-256
1c5d3662fa95d0daf8826582d9cc2a78b3338001b28d2d09a0cb6ddbbdc78f1f
Available editions
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
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. Classical methods and prior constructions are credited. The h₁ baseline is explicit in the cited Kummer source.
Disclosures
AI-assisted tools supported literature discovery, proof development, independent adversarial checks, exact symbolic and rational-arithmetic verification, and manuscript preparation. The author retains responsibility for statements, proofs and citations.
Reproduction
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, with supporting finite diagnostics. Private assessments, internal reviews, correspondence and ongoing inverse-half-power research are excluded.