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Prescribed-gradient minimizers and coefficient stability for homogeneous real-stable polynomials

Scott Schubr

  • Mathematics
  • Combinatorics and Graph Theory
Primary subject
Mathematics
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
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3 · Permanent link
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Overview

Abstract

For each fixed degree n, we prove that among homogeneous real-stable polynomials in n variables with nonnegative coefficients, value one at the all-ones vector, and a prescribed gradient sufficiently close to the all-ones vector, the power of the corresponding linear form uniquely minimizes the squarefree coefficient. This answers Question 1.8 of Gurvits–Leake, arXiv:2609.21083v2. We strengthen the minimum statement to a uniform lower bound for the coefficient deficit by the coefficient ℓ¹ distance to that power. The distance order is optimal. We also classify strict local minimality at every positive gradient and give a cubic example separating local and global minimization. For ternary cubics we obtain the explicit closed gradient radius 1/8 and coefficient-distance constant 1/3584, including singular dispersion. The proof uses centered real-root dispersion, an exact coefficient identity and the classical balanced uniqueness theorem. Outside this cubic case the global neighborhood and constants remain qualitative. We also give a total-variation interpretation for stable count distributions and an illustrative nondeterminantal quartic family.

Keywords real-stable polynomials · prescribed gradient · squarefree coefficient · coefficient stability · hyperbolic polynomials · total variation

Scope of the work

For each fixed degree, prescribed-gradient global uniqueness and coefficient-distance stability hold near the balanced gradient. Ternary cubics have the explicit closed radius 1/8 and coefficient-distance constant 1/3584, including singular dispersion; higher-degree neighborhoods and constants in this paper remain qualitative. Balanced uniqueness is a credited prior input.

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Author disclosures

AI tools assisted exploration, drafting, exact calculations, literature discovery and internal proof checking, as disclosed in the manuscript. Internal analytic audits are not external human refereeing or formal machine certification. Originality remains provisional. Classical balanced rigidity, hyperbolic methods and the author’s published matrix predecessor are credited.

References and related publications

  1. L. Gurvits, Van der Waerden/Schrijver–Valiant like conjectures and stable (aka hyperbolic) homogeneous polynomials: one theorem for all, Electronic Journal of Combinatorics 15 (2008), R66. doi:10.37236/790. arXiv:0711.3496.
  2. L. Gurvits and J. Leake, Unique minimizers for permanents, mixed discriminants, and log-concave polynomials, arXiv:2609.21083v2 (15 October 2026).
  3. L. Gurvits, N. Klein and J. Leake, From trees to polynomials and back again: new capacity bounds with applications to TSP, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024), LIPIcs 297 (2024), 79:1–79:20. doi:10.4230/LIPIcs.ICALP.2024.79. arXiv:2311.09072.
  4. L. Gurvits and J. Leake, Capacity lower bounds via productization, Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC 2021) , 847–858. arXiv:2007.08390.
  5. H. H. Bauschke, O. Güler, A. S. Lewis and H. S. Sendov, Hyperbolic polynomials and convex analysis, Canadian Journal of Mathematics 53 (2001), 470–488. doi:10.4153/CJM-2001-020-6.
  6. P . Brändén, Obstructions to determinantal representability, Advances in Mathematics 226 (2011), 1202–1212. doi:10.1016/j.aim.2010.08.003.
  7. M. Kummer, A note on the hyperbolicity cone of the specialized Vámos polynomial, arXiv:1306.4483v2 (18 August 2015).
  8. S. Schubr, Trace-perturbed minimizers of the mixed discriminant, Luminary Research Archive (2026). Public research record.

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Cite this edition

Scott Schubr. (2026). Prescribed-gradient minimizers and coefficient stability for homogeneous real-stable polynomials (Version 3). Luminary Research Archive. https://research.luminaryoddities.com/prescribed-gradient-minimizers-coefficient-stability-homogeneous-real-stable-polynomials/v/3

BibTeX
@techreport{luminary_prescribed_gradient_minimizers_coefficient_stability_homogeneous_real_stable_polynomials_v3,
  title = {Prescribed-gradient minimizers and coefficient stability for homogeneous real-stable polynomials},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 3},
  url = {https://research.luminaryoddities.com/prescribed-gradient-minimizers-coefficient-stability-homogeneous-real-stable-polynomials/v/3}
}

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