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Research Article · Preprint Mathematics

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

Scott Schubr

Luminary Oddities LLC

  • Mathematics
  • Linear Algebra and Matrix Inequalities
Primary subject
Mathematics
Subject subfield
Linear Algebra and Matrix Inequalities
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
1 · Permanent link
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Overview

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; no new certificate is needed. 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. No claim of optimal constants or an unrestricted-rank conclusion is made.

Keywords permanent · positive semidefinite matrices · Chollet inequality · rank four · Hadamard product · border comparison

Scope of the work

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. No optimal matrix constants, unrestricted-rank Chollet result, new certificate, or worldwide-priority claim is asserted. The existing order-seven Archive edition remains a separate preserved predecessor.

Reproduction notes

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. These checks do not re-prove Rodtes’ theorem or replay Ma’s cited large-order computations. Numerical searches are not proof inputs.

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Author disclosures

AI tools assisted drafting and verification; the author is responsible for the content. Independent internal analytic, attribution, and reader checks are not external human refereeing or formal proof-assistant verification. 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.

References and related publications

  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.

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Chollet's inequality for positive semidefinite matrices of rank at most four (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/chollets-inequality-positive-semidefinite-matrices-rank-at-most-four/v/1

BibTeX
@techreport{luminary_chollets_inequality_positive_semidefinite_matrices_rank_at_most_four_v1,
  title = {Chollet's inequality for positive semidefinite matrices of rank at most four},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/chollets-inequality-positive-semidefinite-matrices-rank-at-most-four/v/1}
}

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