Trace-perturbed minimizers of the mixed discriminant
Loading original first page…
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.
Read the full paper in the original PDF.
Edition files
- Trace_Perturbed_Mixed_Discriminants.pdf61.2 KiB · pdf
SHA-256
bd1fa87c1d44d8b140f396c7d4a1dac25ea45d7735f2a15f384abfdf0ec0d8ee - mixed_discriminant_trace_stability_publication_20261020.zip67.3 KiB · package
SHA-256
d4ed9586c29ea8575e5d4d7bdeaedee8b58344a56a8b55c149510b2c10847eef
Available editions
Research details
- Subjects
- Mathematics · Linear Algebra and Matrix Inequalities
- Keywords
- mixed discriminant, positive semidefinite matrices, Hermitian matrices, trace constraints, stability, constrained Hessian
- Manuscript date
- 2026-10-20
- Reader reuse
- No additional permission granted
Scope
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.
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.
Reproduction
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.
References and related resources
- 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.
- L. Gurvits and J. Leake, Unique minimizers for permanents, mixed discriminants, and log-concave polynomials, arXiv:2609.21083v2, October 15, 2026.
- 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.