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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 ∫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.

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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

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.

Disclosures

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

Reproduction

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.

References and related resources

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