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

Scott Schubr

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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. The proof uses centered real-root dispersion, an exact coefficient identity and the classical balanced uniqueness theorem. The neighborhood and constants are dimension-dependent and qualitative. We also give a total-variation interpretation for stable count distributions and an illustrative nondeterminantal quartic family.

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Edition files

Available editions

  1. Edition 4 · 2026-10-27 · public
  2. Edition 3 · 2026-10-26 · public
  3. Edition 2 · 2026-10-24 · public
  4. Edition 1 · 2026-10-22 · public

Research details

Subjects
Mathematics · Combinatorics and Graph Theory
Keywords
real-stable polynomials, prescribed gradient, squarefree coefficient, coefficient stability, hyperbolic polynomials, total variation
Manuscript date
2026-10-22
Reader reuse
No additional permission granted

Scope

For each fixed degree, global uniqueness and coefficient-ℓ₁ stability hold in a dimension-dependent qualitative neighborhood of the balanced gradient. Strict local minimality is classified at all positive gradients, while a cubic example separates local and global minimization. The balanced uniqueness theorem is a credited prior input.

Disclosures

AI tools assisted exploration, drafting, exact calculations, literature discovery and internal proof checking, as disclosed in the manuscript. Classical balanced rigidity, hyperbolic methods and the author’s published matrix predecessor are credited.

Reproduction

The seven-member package contains the 9-page manuscript, complete source and bibliography, two optional standard-library exact checkers, reader instructions, citation metadata and a manifest. Run verify_quartic.py and verify_local_geometry.py in normal and optimized Python modes. Compile main.tex with Tectonic or two pdflatex runs. The checkers corroborate finite identities and examples; the quantified theorem has an analytic proof.

References and related resources

  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.