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 zᵀLz and zᵀ|R|◦²z, where zᵢ = |xᵢ|², with (per B)(x*B⁻¹x)(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. The cancellation mechanism is classical; the result concerns these auxiliary comparisons.
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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
The note classifies two specific auxiliary cofactor comparisons: sharp constant one in orders one and two, and no finite universal constant at each fixed complex order at least three. It does not disprove Chollet’s conjecture, determine the sign of the full compensated border expression, or decide the separate cofactor matrix comparison. Classical roots-of-unity cancellation is credited.
Disclosures
AI-assisted derivation, exact symbolic checks and separate computational checks were used in developing the note. The theorem is established by its analytic proof. The author is Scott Schubr, Luminary Oddities.
Reproduction
The four-file package contains the complete four-page PDF, editable LaTeX source, reader instructions, and a SHA-256 manifest. The proof is analytic. Compile main.tex with Tectonic or a standard LaTeX installation.
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.