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Research Article · Preprint Mathematics

Sharp limitations of two permanental cofactor comparisons

Scott Schubr

Luminary Oddities

  • Mathematics
  • Linear Algebra and Matrix Inequalities
Primary subject
Mathematics
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
2 · Permanent link
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Overview

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.

Keywords permanent · permanental cofactors · matrix inequalities · correlation matrices · roots of unity · sharp constants

Scope of the work

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.

Reproduction notes

The paper has 4 pages. The editable-source archive contains 4 files. Compile main.tex; the proof is analytic and self-contained.

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Author 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.

References and related publications

  1. B. Reznick. An inequality for products of polynomials. Proceedings of the American Mathematical Society 117 (1993), 1063–1073.
  2. B. Seron, L. Novo and N. J. Cerf. Boson bunching is not maximized by indistinguishable particles. Nature Photonics 17 (2023), 702–709.
  3. 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.

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Sharp limitations of two permanental cofactor comparisons (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/sharp-limitations-two-permanental-cofactor-comparisons/v/2

BibTeX
@techreport{luminary_sharp_limitations_two_permanental_cofactor_comparisons_v2,
  title = {Sharp limitations of two permanental cofactor comparisons},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 2},
  url = {https://research.luminaryoddities.com/sharp-limitations-two-permanental-cofactor-comparisons/v/2}
}

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