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Polynomial Certificates for the Vámos Riesz Obstruction

Scott Schubr

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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 ∫ P^2 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.

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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, signed moment problem, Hankel matrix, polynomial certificates, exact arithmetic, Riesz positivity
Manuscript date
2026-10-25
Reader reuse
No additional permission granted

Scope

The fixed canonical finite signed measure ν=yκ at the specified Vámos specialization is strictly positive on all nonzero ordinary polynomial squares through degree 53; an explicit integer degree 54 square is negative and its 55-dimensional moment matrix has inertia (54,1,0).

Disclosures

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

Reproduction

The 7-page manuscript and editable sources are included in this 9-member package. README.md gives the build and reproduction commands. The standard-library verify.py reconstructs exact branch moments, 55 leading determinant signs and the degree 54 integer witness from certificate.json. verify_independent.py uses the optional python-flint 0.9.0 dependency for formal-Newton and FLINT reconstruction. The manuscript identifies the cited analytic inverse result used to interpret the finite measure.

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.