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Research Article · Preprint Mathematics

An Unrestricted Half-Power Obstruction for the Specialized Vámos Quartic

Scott Schubr

  • Mathematics
  • Analysis and Spectral Theory
Primary subject
Mathematics
Subject subfields
Analysis and Spectral Theory
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
1 · Permanent link
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Overview

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.

Keywords Vámos quartic · complete monotonicity · inverse Laplace transform · hyperbolic polynomials · admissible exponents

Scope of the work

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.

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Author-posted preprint. AI-assisted internal research and verification do not constitute external human peer review or proof-assistant formalization.

References and related publications

  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.
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  10. A. H. Zemanian, Inversion formulas for the distributional Laplace transformation, SIAM J. Appl. Math. 14 (1966), 159–166. doi:10.1137/0114013.

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Scott Schubr. (2026). An Unrestricted Half-Power Obstruction for the Specialized Vámos Quartic (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/unrestricted-half-power-obstruction-specialized-vamos-quartic/v/1

BibTeX
@techreport{luminary_unrestricted_half_power_obstruction_specialized_vamos_quartic_v1,
  title = {An Unrestricted Half-Power Obstruction for the Specialized Vámos Quartic},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/unrestricted-half-power-obstruction-specialized-vamos-quartic/v/1}
}

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