Effective Prescribed-Gradient Stability for Homogeneous Real-Stable Polynomials
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Abstract
For homogeneous real-stable polynomials of degree n in n variables, with nonnegative coefficients and value one at the all-ones vector, we give explicit positive rational constants in every fixed degree for a global prescribed-gradient coefficient reserve. On an explicitly specified closed neighborhood of the balanced gradient, the squarefree coefficient deficit dominates both centered-root dispersion and the coefficient ℓ^1 distance from the corresponding power of a linear form. The proof replaces the compactness step in the earlier qualitative theorem by a radial extension to a degree-drop boundary and a positive contraction dichotomy. An explicit positive map then corrects the gradient while preserving the centered factors. The bounds include singular dispersion and stable coefficient boundaries. We also obtain an explicit Frobenius-distance reserve for Hermitian mixed discriminants. The constants are deliberately coarse.
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- Effective_stable_polynomial_reserves.pdf82.2 KiB · pdf
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Research details
- Subjects
- Mathematics · Combinatorics
- Keywords
- real-stable polynomials, prescribed gradient, coefficient stability, mixed discriminants, hyperbolic polynomials
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
Explicit positive rational fixed-degree reserves control dispersion and coefficient distance for normalized homogeneous real-stable polynomials near balanced gradients, including singular and boundary cases, with mixed-discriminant consequences.
Disclosures
AI tools assisted with exploration, drafting, literature discovery, exact calculations and internal checking. Internal checks are not external human refereeing or formal proof-assistant verification.
Reproduction
The 7-page manuscript and editable LaTeX sources are supplied in this 7-member package. README.txt gives build instructions. verify_reserve.py provide the finite mathematical controls specified in the manuscript alongside its analytic proofs.