Trace-perturbed minimizers of the mixed discriminant
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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.
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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 every fixed integer n >= 1, a sufficiently small neighborhood of the uniform trace vector is considered, with sum t_i = n, ||t - 1||_1 < epsilon_n, complex Hermitian positive semidefinite n-tuples, Tr X_i = t_i, and sum X_i = I. The unique minimizer is X_i = (t_i/n)I, with normalized minimum prod(t_i)/n^n. The exact Hessian criterion describes the local second variation.
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. AI tools assisted with derivations, literature searches, proof checking, and manuscript preparation.
Reproduction
Manuscript version 1.0, October 20, 2026; first public Archive edition 1. The publication package contains the complete four-page PDF, self-contained LaTeX source with embedded bibliography, reader build instructions, BibTeX entries, and a checksum manifest for the other four files. The proof is analytic. Standard LaTeX packages are required; the README gives Tectonic and pdfLaTeX instructions. Rebuilt PDFs may differ in bytes because of compiler, font, or metadata differences.
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.