Sharp limitations of two permanental cofactor comparisons
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Abstract
We study two auxiliary cofactor comparisons motivated by permanent Hadamard inequalities. For a complex positive definite correlation matrix B, let R be its transposed permanental cofactor matrix and L that of its entrywise squared modulus. We compare z^TLz and z^T|R|^( ∘ 2)z, where z_i=|x_i|^2, with (per B)(x^*B^(-1)x)(x^*Rx). Both comparisons have sharp universal constant one in orders one and two. At every fixed order at least three, neither admits any finite universal constant, even among dense irreducible positive definite correlation matrices with nonreal triangle phases. An explicit roots-of-unity family gives quantitative divergence.
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Research details
- Subjects
- Mathematics · Linear Algebra and Matrix Inequalities
- Keywords
- permanent, permanental cofactors, matrix inequalities, correlation matrices, roots of unity, sharp constants
- Manuscript date
- 2026-10-22
- Reader reuse
- No additional permission granted
Scope
Sharp classification of two auxiliary permanental cofactor comparisons: constant one in orders one and two, and unbounded ratios at each fixed order at least three even for the stated dense positive definite complex correlation matrices.
Disclosures
AI-assisted derivation, exact symbolic checks and separate computational checks were used in developing the note. The theorem is analytic and its proof does not depend on numerical search or finite-sample inference. Internal scientific and reader checks do not constitute external human refereeing. The author is Scott Schubr, Luminary Oddities LLC.
Reproduction
The paper has 4 pages. The editable-source archive contains 4 files. Compile main.tex; the proof is analytic and self-contained.
References and related resources
- B. Reznick. An inequality for products of polynomials. Proceedings of the American Mathematical Society 117 (1993), 1063–1073.
- B. Seron, L. Novo and N. J. Cerf. Boson bunching is not maximized by indistinguishable particles. Nature Photonics 17 (2023), 702–709.
- L. Pioge, K. K. Pietrasz, B. Seron, L. Novo and N. J. Cerf. A logical implication between two conjectures on matrix permanents. Linear Algebra and its Applications 725 (2025), 309–318.