Research Article · Preprint Mathematics
Effective Prescribed-Gradient Stability for Homogeneous Real-Stable Polynomials
- Primary subject
- Mathematics
- Subject subfields
- Combinatorics
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
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.
Keywords real-stable polynomials · prescribed gradient · coefficient stability · mixed discriminants · hyperbolic polynomials
Scope of the work
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.
Reproduction notes
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.
The manuscript
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Cite this edition
Scott Schubr. (2026). Effective Prescribed-Gradient Stability for Homogeneous Real-Stable Polynomials (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/effective-coefficient-reserves-homogeneous-real-stable-polynomials/v/2
BibTeX
@techreport{luminary_effective_coefficient_reserves_homogeneous_real_stable_polynomials_v2,
title = {Effective Prescribed-Gradient Stability for Homogeneous Real-Stable Polynomials},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/effective-coefficient-reserves-homogeneous-real-stable-polynomials/v/2}
}Loading discussion…