Prescribed-gradient minimizers and coefficient stability for homogeneous real-stable polynomials
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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. 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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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
The theorem concerns homogeneous degree-n real-stable polynomials in n variables with nonnegative coefficients, normalized at the all-ones vector, for every fixed dimension and a sufficiently small prescribed-gradient neighborhood. The neighborhood and constants are qualitative and dimension-dependent. Classical balanced uniqueness is a cited prior input. This extends the author’s matrix predecessor. The probability formulation translates the same theorem.
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 six-member publication package contains the complete seven-page PDF, main.tex with bibliography, reader instructions, a citation entry, an optional Python exact-arithmetic checker and a five-file SHA-256 manifest. Compile main.tex with Tectonic or run pdflatex twice. The optional standard-library checker reproduces two illustrative quartic identities; the general proof is analytic.
References and related resources
- L. Gurvits, Van der Waerden/Schrijver–Valiant like conjectures and stable homogeneous polynomials (2008)
- L. Gurvits and J. Leake, Unique minimizers for permanents, mixed discriminants, and log-concave polynomials, arXiv:2609.21083v2
- L. Gurvits, N. Klein and J. Leake, From trees to polynomials and back again (ICALP 2024)
- L. Gurvits and J. Leake, Capacity lower bounds via productization (2021)
- H. H. Bauschke, O. Güler, A. S. Lewis and H. S. Sendov, Hyperbolic polynomials and convex analysis (2001)
- P. Brändén, Obstructions to determinantal representability (2011)
- M. Kummer, A note on the hyperbolicity cone of the specialized Vámos polynomial
- S. Schubr, Trace-perturbed minimizers of the mixed discriminant (2026)