Effective coefficient reserves 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 ℓ¹ 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.pdf81.8 KiB · pdf
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214aaa054a82de2d156769dc7c45ea419cb4d2e05fc82b3e467c882d80c7215f - Effective_stable_reserve_publication.zip93.6 KiB · package
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89b224bbb611cff8ec295c52e2a52d27a3d499437020a258aaba826a1e09dcd0
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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 degree-dependent rational reserves for normalized homogeneous nonnegative real-stable polynomials near the balanced gradient, including boundary and Hermitian mixed-discriminant cases.
Disclosures
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Reproduction
Editable LaTeX source, build helper and exact standard-library checker are included. With Python 3.10 or later, run python verify_reserve.py and python -O verify_reserve.py. The finite checks corroborate normalizations and selected identities; the universal theorem is analytic. Build the PDF with the supplied Tectonic helper or a current LaTeX installation.