Strong permanent inequalities for three-row bipartite supports and triangle-free supports through order seven
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Abstract
For every complex 3 × q contraction X, we prove that the permanent of the block matrix with diagonal blocks I₃ and I_q and off-diagonal blocks X and X* is at least the permanent of its entrywise squared modulus. The proof retains all square-minor orders and controls the coupled third-order cancellations through the row Gram matrix. It gives an explicit sum of nonnegative lower bounds and includes arbitrary phases, zero entries and singular endpoints. Combining this theorem with three pentagon blow-up cases and an exact finite graph classification proves the same strong inequality for every positive semidefinite matrix with triangle-free support on at most seven vertices, after diagonal scaling. Two previously unpublished blow-up cases are proved here, one analytically and one by an explicit positive integer polynomial certificate. Reproducibility supplements contain the certificate and finite classification verifier. The unrestricted Chollet conjecture is not settled.
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Research details
- Subjects
- Mathematics · Linear Algebra and Matrix Inequalities
- Keywords
- permanent inequalities, positive semidefinite matrices, three-row contractions, triangle-free graphs, Chollet conjecture, polynomial certificates
- Manuscript date
- 2026-10-22
- Reader reuse
- No additional permission granted
Scope
Analytic strong permanent contraction on every finite K₃,q support and strong comparison on all triangle-free PSD supports through order seven, with explicit nonbipartite certificate and finite host classification.
Disclosures
AI tools assisted with drafting and verification. The author is responsible for the statements, proofs, attribution and final text. This author-published preprint has not undergone formal journal peer review.
Reproduction
The paper has 8 pages. The editable-source archive contains 9 files. Compile manuscript/main.tex. Standard-library Python verifies the 119-term nonadjacent certificate and enumerates the triangle-free seven-vertex graph hosts.
References and related resources
- 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.
- P. Pant and R. Singh, Structural Classes for Chollet's Permanent Conjecture, arXiv:2604.24192v2 (2026).
- S. Schubr, Strong permanent inequalities for complex positive semidefinite cycle and single-chord supports, Luminary Research Archive, first edition (2026).
- S. Schubr, Strong permanent inequalities for complex two- and three-chord seven-cycle supports, Luminary Research Archive, first edition (2026).
- R. Mehatari, M. R. Kannan and A. Samanta, On the adjacency matrix of a complex unit gain graph, Linear and Multilinear Algebra 70 (2022), no. 9, 1798–1813.
- D. Handelman, Representing polynomials by positive linear functions on compact convex polyhedra, Pacific Journal of Mathematics 132 (1988), no. 1, 35–62.