Strong permanent inequalities for three-row bipartite and triangle-free supports
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Abstract
For every complex 3 × q contraction X, we prove that the permanent of [I₃ X; X* Iq] 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 Two previously unpublished blow-up cases are proved here, one analytically and one by an explicit positive integer polynomial certificate.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.
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- Strong_Permanent_Three_Row_Triangle_Free.pdf105.0 KiB · pdf
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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
Proves the strong permanent inequality for complex K₃,q positive semidefinite supports, with the full contraction hypothesis, and for triangle-free positive semidefinite supports on at most seven vertices. Includes arbitrary phases, zero entries and singular endpoints. Uses the previously published three-chord theorem for the adjacent-doubled pentagon case.
Disclosures
AI tools assisted with drafting and verification. The author is responsible for the statements, proofs, attribution and final text.
Reproduction
The complete publication package includes the eight-page manuscript and LaTeX source, a 119-term positive integer polynomial certificate, its exact rational verifier, and an exhaustive triangle-free graph classification verifier. The executable supplements require only the Python standard library. The three-row theorem is analytic; finite verification concerns the stated certificate and graph enumeration.
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.