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An Unrestricted Half-Power Obstruction for the Specialized Vámos Quartic

Scott Schubr

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Abstract

The inverse square root of the specialized four-variable Vámos quartic is shown not to be completely monotone. More generally, the obstruction holds for every interior stable member hλ = 4e₁e₃ + λa²b² with 0 < λ < 64/27. The proof computes a regular correction to a singular positive leading inverse, establishes distributional Laplace-inversion convergence through uniform complex estimates, and identifies a negative density outside the leading support. A smooth-section argument transfers this sign to the unrestricted inverse. For the original quartic an explicit interval and uniform negative bound are obtained. The result gives a qualitative strict exponent gap above 1/2, uniform on compact interior parameter sets. At λ = 64/27 the first correction is nonnegative outside the leading support; complete monotonicity at that endpoint remains open.

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Edition files

Available editions

  1. Edition 5 · 2026-10-31 · public
  2. Edition 4 · 2026-10-29 · public
  3. Edition 3 · 2026-10-27 · public
  4. Edition 2 · 2026-10-25 · public
  5. Edition 1 · 2026-10-23 · public

Research details

Subjects
Mathematics · Analysis and Spectral Theory
Keywords
Vámos quartic, complete monotonicity, inverse Laplace transform, hyperbolic polynomials, admissible exponents
Manuscript date
2026-10-22
Reader reuse
No additional permission granted

Scope

Four-variable inverse-half-power obstruction for every interior stable parameter, with a qualitative exponent gap. Endpoint positivity at 64/27, all admissible exponents and a general large-exponent conjecture remain unresolved.

Disclosures

Author-posted preprint. AI-assisted internal research and verification were used.

Reproduction

The six-member package contains main.pdf, main.tex, verify.py, reader instructions, citation metadata and checksums. python -B verify.py and python -O -B verify.py use only the standard library and report 30 named algebraic checks. These do not replace the complex domination, distributional convergence or positivity arguments in the manuscript.

References and related resources

  1. P. Brändén, Obstructions to determinantal representability, Adv. Math. 226 (2011), 1202–1212. doi:10.1016/j.aim.2010.08.003.
  2. M. Kummer, A note on the hyperbolicity cone of the specialized Vámos polynomial, Acta Appl. Math. 144 (2016), 11–15. doi:10.1007/s10440-015-0036-z.
  3. Kh. Kozhasov, M. Michałek and B. Sturmfels, Positivity certificates via integral representations, in Facets of Algebraic Geometry , Vol. 2, Cambridge University Press, 2022, pp. 84–114. doi:10.1017/9781108877855.004. arXiv:1908.04191.
  4. E. Mazzucchelli and P. Raman, Canonical forms as dual volumes, preprint (2025), arXiv:2509.02239v1.
  5. M. Michałek, B. Sturmfels, C. Uhler and P. Zwiernik, Exponential varieties, Proc. Lond. Math. Soc. (3) 112 (2016), 27–56. doi:10.1112/plms/pdv066.
  6. F. W. J. Olver et al. (eds.), NIST Digital Library of Mathematical Functions , https://dlmf. nist.gov/, Sections 10.29–10.32 and 10.43, accessed October 22, 2026.
  7. S. Schubr, Sharp stability and geometric classification of a Vámos quartic deformation, Luminary Research Archive, preprint, edition 1 (2026), Theorem 2.2. Public edition .
  8. A. D. Scott and A. D. Sokal, Complete monotonicity for inverse powers of some combinatorially defined polynomials, Acta Math. 213 (2014), 323–392. doi:10.1007/s11511-014-0121-6.
  9. D. Wei, Riesz kernels of hyperbolic polynomials: positivity, admissible exponents and Jordan rigidity, preprint (2026), arXiv:2609.31592v2.
  10. A. H. Zemanian, Inversion formulas for the distributional Laplace transformation, SIAM J. Appl. Math. 14 (1966), 159–166. doi:10.1137/0114013.