Row-dependent bounds for strong permanent contraction on four-by-eight supports
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Abstract
We prove that every complex positive semidefinite matrix A supported in K₄,₈ satisfies per(|A|²) ≤ per(A) ∏ᵢ aᵢᵢ. The proof extends an established four-row framework by retaining a necessary lower bound on squared row moments and choosing a convex permanent bound as a polynomial of the row norms. An exact two-point obstruction shows why one fixed mixture in the inherited scalar framework is insufficient. A small-row reduction and a rational tensor Bernstein certificate cover the capped-entry region; a published all-column large-entry theorem covers its complement. The result is computer-assisted and includes arbitrary phases and singular endpoints within the stated supports.
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- Strong_Permanent_K48.pdf112.6 KiB · pdf
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6dbabfd38969b7d0a11fc83526e9ec40c10ee6cde47e825ea62c93ba2575a0a4 - K48_Publication_Package.zip123.2 KiB · package
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Research details
- Subjects
- Mathematics · Linear Algebra and Matrix Inequalities
- Keywords
- permanent inequalities, Chollet conjecture, positive semidefinite matrices, bipartite supports, Bernstein certificates, computer-assisted proof
- Manuscript date
- 2026-10-22
- Reader reuse
- No additional permission granted
Scope
Strong Q(B) ≤ d(B)P(B) for Hermitian positive semidefinite matrices supported in K₄,₈, including complex phases, missing edges, zero rows, and singular endpoints. A continuation of the published four-row results through six and seven columns.
Disclosures
AI tools assisted mathematical research, drafting, and verification. The author is responsible for the statements, proofs, attribution, and final text.
Reproduction
The publication package contains the manuscript PDF, editable LaTeX source, a standalone Python exact-rational certificate verifier, reader instructions, citation metadata, and checksums. Run python -B verify_k48_certificate.py with Python 3.10 or newer. The verifier reconstructs 720 small-row corner/pairing values and 65,536 Bernstein coefficients across 256 boxes. The manuscript supplies the analytic reductions and the cited large-entry input.
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.
- E. Carlen, E. H. Lieb and M. Loss. An inequality of Hadamard type for permanents. Methods and Applications of Analysis 13 (2006), no. 1.
- P. Pant and R. Singh. Structural Classes for Chollet’s Permanent Conjecture. arXiv:2604.24192v2 (2026).
- 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.
- Z. Zeng, H. Liu and K. Ratnavelu. Sharp Self-Chollet Inequalities in Order Five: Rank Two and Structured Rank Three. Preprints.org, version 1 (2026), not peer-reviewed.
- 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).