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Polynomial Certificates for the Vámos Riesz Obstruction

Scott Schubr

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

Abstract

We determine the first ordinary polynomial-square degree that detects the negative sign of a canonical correction measure associated with the specialized four-variable Vámos quartic. If κ is the compact finite-part correction from the earlier inverse-Laplace analysis and ν = yκ, then ∫P² dν > 0 for every nonzero real polynomial of degree at most 53, whereas an explicit integer polynomial of degree 54 has negative square integral. Equivalently, the moment matrix through degree 53 is positive definite and the next matrix has exactly one negative eigenvalue. The proof converts one algebraic analytic branch into rational moments and uses exact fraction-free elimination. A portable standard-library verifier reconstructs the certificate; a second implementation recovers the moments from an implicit quartic equation and independently checks the determinants and polynomial. The earlier analytic paper supplies the identification of the correction and the full inverse obstruction. The finite theorem here does not specify a mixed-derivative violation of the original quartic inverse.

Keywords Vámos quartic · signed moment problem · Hankel matrix · polynomial certificates · exact arithmetic · Riesz positivity

Scope of the work

At λ = 1, the correction measure ν = yκ has positive polynomial-square integrals through degree 53 and an explicit integer negative-square certificate of degree 54; the degree-54 Hankel matrix has inertia (54, 1, 0). The analytic measure identification is inherited from the cited predecessor. No finite four-variable derivative counterexample, effective small-parameter bound, endpoint closure or theta result is claimed.

Reproduction notes

The nine-member package supplies the seven-page manuscript, editable LaTeX source, all 55 integer polynomial coefficients, exact square value, two portable checkers, pinned optional python-flint dependency, reader instructions, citation metadata and SHA256SUMS.txt. The standard-library checker and the independent implicit-quartic/FLINT checker reconstruct the moments and verify all 55 leading determinant signs, 54 orthogonality identities and the negative square value. Both pass normally and under Python -O. The analytic Cauchy-transform identification is cited and is separate from the finite arithmetic checks.

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Author disclosures

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.
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  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 .
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  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.

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Cite this edition

Scott Schubr. (2026). Polynomial Certificates for the Vámos Riesz Obstruction (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/unrestricted-half-power-obstruction-specialized-vamos-quartic/v/2

BibTeX
@techreport{luminary_unrestricted_half_power_obstruction_specialized_vamos_quartic_v2,
  title = {Polynomial Certificates for the Vámos Riesz Obstruction},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 2},
  url = {https://research.luminaryoddities.com/unrestricted-half-power-obstruction-specialized-vamos-quartic/v/2}
}

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