Prescribed-Gradient Coefficient Stability for Lorentzian Polynomials
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Abstract
For normalized Lorentzian polynomials of degree n in n variables, we prove that the power of a linear form with prescribed gradient is the unique minimizer of the squarefree coefficient whenever that gradient is sufficiently close to the all-ones vector. The coefficient excess controls the full coefficient ℓ¹ distance, uniformly in each fixed degree. The proof gives an explicit separation bound for the cone determined by the transverse quadratic derivatives at the balanced power. This bound also yields an explicit local stability estimate when both gradient and polynomial are close to that power. Variable splitting extends the result to positive integer compositions, and volume polynomials give a mixed-volume coefficient stability bound.
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Edition files
- Lorentzian_coefficient_stability.pdf313.7 KiB · pdf
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b1f690ea9181b94e00da284e1d2392d64a991c2492d4bafdfc0954eedb55ebe6 - Lorentzian_source_package.zip319.9 KiB · package
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e44e3ccba8bdc09d8defd6db2f563acd75469ddb780cd3919a515fc8d88d2d0a
Available editions
- Edition 1 · 2026-11-01 · public
Research details
- Subjects
- Mathematics · Combinatorics and Graph Theory · Geometry and Optimal Transport
- Keywords
- Lorentzian polynomials, prescribed gradients, coefficient stability, mixed volumes
- Manuscript date
- 2026-11-01
- Reader reuse
- none
Scope
Fixed-degree prescribed-gradient coefficient stability for normalized Lorentzian polynomials near the balanced gradient, with explicit derivative-cone and joint local bounds and composition/mixed-volume recipients.
Disclosures
AI tools assisted with mathematical exploration, literature discovery, drafting and internal checking. The author is responsible for the manuscript.
Reproduction
The source package contains the complete editable manuscript and build instructions. The analytic proofs appear in the article. The portable rational-control program supplements the analytic proofs.