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Chollet's inequality for positive semidefinite matrices of rank at most four

Scott Schubr

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Abstract

We prove quantitative self-conjugate Chollet inequalities for all complex Hermitian positive semidefinite matrices of rank at most four in orders eight and nine, with respective factors 243/320 and 1215/1352. A short extension of the occupation-border comparison of Li, Zhao and Luo balances its two multiplicative losses. The input is the previously established singular order-five factor 3/5. Together with Rodtes’ order-four theorem and Ma’s theorem for orders at least ten, the new estimates imply Chollet’s inequality in every order when both factors have rank at most four. The coefficient-one inequality is strict for positive diagonal in orders at least five.

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Edition files

Available editions

  1. Edition 4 · 2026-11-02 · public
  2. Edition 3 · 2026-10-29 · public
  3. Edition 2 · 2026-10-27 · public
  4. Edition 1 · 2026-10-22 · public

Research details

Subjects
Mathematics · Linear Algebra and Matrix Inequalities
Keywords
permanent, positive semidefinite matrices, Chollet inequality, rank four, Hadamard product, border comparison
Manuscript date
2026-10-22
Reader reuse
No additional permission granted

Scope

The new full-domain estimates concern complex Hermitian positive semidefinite matrices of rank at most four in orders eight and nine; the order-seven coefficient also improves. The all-orders paired conclusion requires both factors to have rank at most four and combines these bounds with the cited singular-five base, Rodtes’ low-order theorem, and Ma’s theorem for orders at least ten. The existing order-seven Archive edition remains a separate preserved predecessor.

Disclosures

AI tools assisted drafting and verification; the author is responsible for the content. The occupation-border identities are credited to Li, Zhao and Luo; the singular-five certificate is reused from the author’s earlier paper and its feature framework follows Lavi. The all-orders conclusion explicitly cites Rodtes and Ma.

Reproduction

The seven-member package contains the complete five-page PDF, editable LaTeX source, the unchanged inherited singular-five certificate and exact-integer verifier, finite scalar comparison code, reader instructions, and a six-file SHA-256 manifest. The two Python verifiers use the standard library. The certificate verifier reconstructs 6,210 coefficient identities and checks 38 positive-definiteness witnesses. The package checks the stated finite certificate and scalar comparisons.

References and related resources

  1. J. Chollet, Is there a permanental analogue to Oppenheim’s inequality?, American Mathematical Monthly 89 (1982), 57–58.
  2. S. Schubr, A quantitative Chollet inequality for rank-four matrices of order seven, Luminary Research Archive, edition 1 (20 October 2026).
  3. K. Rodtes, Chollet’s permanent conjecture for 4×4 matrices, Linear and Multilinear Algebra 72 (2024), 2633–2638.
  4. Y. Ma, Quantitative Chollet Inequalities for Matrices of Rank at Most Four: Spectral bounds and an order-nine tight-frame construction, arXiv:2609.37461v1 (28 September 2026).
  5. Q. Li, Q. Zhao and W. Luo, Chollet’s Permanent Conjecture Through Order Six via Border Induction, Preprints.org 202608.2196v1 (31 August 2026).
  6. Y. Lavi, A Conjecture on the Permanent of Positive Semidefinite Matrices, preprint dated 26 September 2026, Zenodo record posted 27 September 2026.