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Row-dependent bounds for strong permanent contraction on four-by-eight supports

Scott Schubr

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Abstract

We prove that every complex positive semidefinite matrix A supported in K_(4,8) satisfies per(|A|^2)≤per(A)∏_i a_(ii). 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.

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Edition files

Available editions

  1. Edition 2 · 2026-10-27 · public
  2. Edition 1 · 2026-10-23 · public

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

Computer-assisted strong permanent contraction on every complex PSD K₄,₈ support, using a row-dependent convex bound and an exact Bernstein certificate together with the cited large-entry reduction.

Disclosures

AI tools assisted mathematical research, drafting, and verification. The author is responsible for the statements, proofs, attribution, and final text. Archive posting and internal checks do not constitute external peer review.

Reproduction

The paper has 8 pages. The editable-source archive contains 7 files. Compile manuscript/main.tex. The standard-library verify_k48_certificate.py reconstructs the fixed-mixture obstruction, small-row controls and 65,536 Bernstein coefficients.

References and related resources

  1. L. B. Beasley. An inequality on permanents of Hadamard products. Bulletin of the Korean Mathematical Society 37 (2000), no. 3, 633–639.
  2. J. Chollet. Is there a permanental analogue to Oppenheim’s inequality? American Mathematical Monthly 89 (1982), 57–58.
  3. S. W. Drury. A counterexample to a question of Bapat and Sunder. Electronic Journal of Linear Algebra 31 (2016), 69–70.
  4. E. Carlen, E. H. Lieb and M. Loss. An inequality of Hadamard type for permanents. Methods and Applications of Analysis 13 (2006), no. 1.
  5. P. Pant and R. Singh. Structural Classes for Chollet’s Permanent Conjecture. arXiv:2604.24192v2 (2026).
  6. 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.
  7. 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.
  8. S. Schubr. Strong permanent inequalities for three-row bipartite and triangle-free supports. Luminary Research Archive, first edition (2026).
  9. S. Schubr. Strong permanent contraction on four-row bipartite supports through six columns. Luminary Research Archive, first edition (2026).
  10. S. Schubr. Strong permanent contraction on four-by-seven bipartite supports. Luminary Research Archive, first edition (2026).