Strong Self-Permanent Inequalities for Complex Four-Row Bipartite Supports
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Abstract
The paper proves a strong self-conjugate permanent inequality for every complex positive semidefinite matrix whose off-diagonal support is bipartite with one part of size at most four. The permanent of its entrywise squared modulus is bounded by its permanent times the product of its diagonal entries. The other part has arbitrary finite size, so the result permits unbounded matrix order and arbitrary complex cycle phases. The proof combines a matching-polytope overlap bound with a column-independent endpoint argument and a large-entry reduction. A five-by-five extension, proved by an exact polynomial certificate, gives the same strong inequality for every bipartite support through total order ten. A separate total-energy estimate gives the ordinary self-Chollet inequality on every positive semidefinite bipartite support with one part of size at most twelve. Complete analytic reductions and portable exact verification are supplied.
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Research details
- Subjects
- Mathematics · Linear Algebra and Matrix Inequalities
- Keywords
- permanent inequalities, positive semidefinite matrices, bipartite supports, Chollet conjecture, Bernstein certificates
- Manuscript date
- 2026-10-28
- Reader reuse
- none
Scope
Strong self-permanent comparison for complex positive semidefinite K₄,q supports with every finite q; a K₅,₅ exact-certificate extension; an ordinary self-Chollet energy criterion including K₁₂,q; and specified five-row regimes.
Disclosures
Generative AI tools assisted mathematical exploration, proof development, drafting, verification-code development and internal checking. OpenAI Codex (GPT-6.1 Sol; gpt-6.1-sol) assisted the journal-presentation revision. Scott Schubr directed this work and assumes responsibility for its content, mathematical arguments and attribution. The work was self-funded.
Reproduction
The complete manuscript has 17 pages. The seven-member source package contains the PDF, editable LaTeX, exact verification code, build helper, reader instructions, citation metadata and checksum manifest. Compile with pdfLaTeX using the complete official SIAM 2016 macro distribution as an external dependency. The standard-library Python checker reconstructs all 3,145,728 exact Bernstein coefficients.
References and related resources
- L. B. Beasley, An inequality on permanents of Hadamard products, Bulletin of the Korean Mathematical Society 37 (2000), no. 3, 633–639.
- J. Chollet, Is there a permanental analogue to Oppenheim’s inequality?, American Mathematical Monthly 89 (1982), 57–58.
- S. W. Drury, A counterexample to a question of Bapat and Sunder, Electronic Journal of Linear Algebra 31 (2016), 69–70. https://journals.uwyo.edu/index.php/ela/article/view/1631.
- E. Carlen, E. H. Lieb and M. Loss, An inequality of Hadamard type for permanents, Methods and Applications of Analysis 13 (2006), no. 1. https://doi.org/10.4310/MAA.2006.v13.n1.a1. Square form: arXiv:math/0508096v1, Theorem 1.1, https://arxiv.org/html/math/0508096v1.
- E. H. Lieb, Proofs of some conjectures on permanents, Journal of Mathematics and Mechanics 16 (1966), 127–134.
- J. Edmonds, Maximum matching and a polyhedron with 0,1-vertices, Journal of Research of the National Bureau of Standards, Section B 69B (1965), 125–130. https://doi.org/10.6028/jres.069B.013.
- P. Pant and R. Singh, Structural Classes for Chollet’s Permanent Conjecture, arXiv:2604.24192v2 (2026). https://arxiv.org/abs/2604.24192v2.
- S. Sa-nguansin and K. Rodtes, Permanents of correlation matrices and the Chollet permanental conjecture, Linear and Multilinear Algebra 73 (2025), no. 18, 4084–4096. https://doi.org/10.1080/03081087.2025.2588584.
- S. Schubr, Strong permanent inequalities for three-row bipartite and triangle-free supports, Luminary Research Archive, first edition (2026).
- S. Schubr, Strong permanent contraction on four-row bipartite supports through six columns, Luminary Research Archive, first edition (2026).
- S. Schubr, Strong permanent contraction on four-by-seven bipartite supports, Luminary Research Archive, first edition (2026).
- S. Schubr, Row-dependent bounds for strong permanent contraction on four-by-eight supports, Luminary Research Archive, first edition (2026).
- Z. Zeng, H. Liu and K. Ratnavelu, Sharp Self-Chollet Inequalities in Order Five: Rank Two and Structured Rank Three, Preprints.org (2026), version 1, posted 18 August 2026, Section 6. https://www.preprints.org/manuscript/202608.1240/v1.
- Q. Li, Q. Zhao and W. Luo, Chollet’s Permanent Conjecture Through Order Six via Border Induction, Preprints.org (2026), version 1, posted 31 August 2026.