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Trace-perturbed minimizers of the mixed discriminant

Scott Schubr

  • 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

For each fixed matrix size, we prove that the scalar tuple uniquely minimizes the mixed discriminant of positive semidefinite Hermitian matrices whose sum is the identity and whose individual traces are sufficiently close to one. This answers Question 1.7 of Gurvits--Leake, arXiv:2609.21083v2. The argument combines Gurvits's doubly stochastic uniqueness theorem with a positive constrained Hessian and compactness. We also record the exact region of positive definiteness of that Hessian at a scalar tuple. The global neighborhood in the theorem is qualitative and may depend on the size.

Keywords mixed discriminant · positive semidefinite matrices · Hermitian matrices · trace constraints · stability · constrained Hessian

Scope of the work

For each fixed matrix size, sufficiently small individual-trace perturbations preserve the unique scalar-tuple mixed-discriminant minimizer among Hermitian PSD tuples summing to the identity. The constrained scalar-tuple Hessian positive-definiteness region is determined exactly.

Reproduction notes

The 4-page analytic note, complete main.tex, CITATION.bib and README.txt are supplied with MANIFEST.sha256.json in this 5-member package. README.txt gives standard Tectonic/pdfLaTeX build instructions and specifies the cited preprint versions. No computational certificates or datasets are required.

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Author disclosures

This specialist note relies on Gurvits's earlier doubly stochastic global uniqueness theorem and standard stability and compactness arguments. The close argument in his Corollary 4.2 and the already-known simultaneously diagonalizable permanent subcase are explicitly credited. External novelty remains provisional; no worldwide priority or practical application is claimed. AI tools assisted with derivations, literature searches, proof checking, and manuscript preparation. Internal AI-assisted checks do not constitute external human refereeing or formal verification.

References and related publications

  1. L. Gurvits, Van der Waerden conjecture for mixed discriminants, Advances in Mathematics 200 (2006), no. 2, 435-454. DOI 10.1016/j.aim.2004.12.002. Cited statement numbering follows arXiv:math/0406420v1, June 21, 2004.
  2. L. Gurvits and J. Leake, Unique minimizers for permanents, mixed discriminants, and log-concave polynomials, arXiv:2609.21083v2, October 15, 2026.
  3. L. Gurvits, N. Klein, and J. Leake, From trees to polynomials and back again: New capacity bounds with applications to TSP, arXiv:2311.09072v2, May 9, 2024.

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Trace-perturbed minimizers of the mixed discriminant (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/trace-perturbed-minimizers-of-the-mixed-discriminant/v/2

BibTeX
@techreport{luminary_trace_perturbed_minimizers_of_the_mixed_discriminant_v2,
  title = {Trace-perturbed minimizers of the mixed discriminant},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 2},
  url = {https://research.luminaryoddities.com/trace-perturbed-minimizers-of-the-mixed-discriminant/v/2}
}

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