PreprintarXiv licencearXiv
Bounded Ratios of Lorentzian Polynomials II: The Complete Quadratic Local-to-Global Classification
Abstract
Every quadratic Hessian slice of a Lorentzian polynomial yields bounded monomial ratios among the normalized coefficients of the polynomial. We determine exactly for which pairs (n,d) these quadratic-slice ratios generate the full bounded-ratio cone for every M-convex support S⊆Δnd. For d≥ 2, this quadratic local-to-global principle holds universally if and only if n≤ 3, d=2, or (n,d)=(4,3). In every remaining case, the principle fails already for Lorentzian polynomials with full support: for cubics in n≥ 5 variables and for polynomials of degree d≥ 4 in n≥ 4 variables. We identify the two minimal obstructions, at (n,d)=(4,4) and (n,d)=(5,3), and propagate them throughout the failure region by degree and variable aggregation.
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BibTeX
@misc{chen2026bounded,
title = {{Bounded Ratios of Lorentzian Polynomials II: The Complete Quadratic Local-to-Global Classification}},
author = {Dijia Chen and Bowen Gan and Ivy Liu and Zemeng Wang and Chengzhi Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07098},
eprint = {2609.07098},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07098},
}
RIS
TY - UNPB
TI - Bounded Ratios of Lorentzian Polynomials II: The Complete Quadratic Local-to-Global Classification
AU - Dijia Chen
AU - Bowen Gan
AU - Ivy Liu
AU - Zemeng Wang
AU - Chengzhi Wu
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.07098
UR - https://arxiv.org/abs/2609.07098
AB - Every quadratic Hessian slice of a Lorentzian polynomial yields bounded monomial ratios among the normalized coefficients of the polynomial. We determine exactly for which pairs (n,d) these quadratic-slice ratios generate the full bounded-ratio cone for every M-convex support S⊆Δ_n^d. For d≥ 2, this quadratic local-to-global principle holds universally if and only if n≤ 3, d=2, or (n,d)=(4,3). In every remaining case, the principle fails already for Lorentzian polynomials with full support: for cubics in n≥ 5 variables and for polynomials of degree d≥ 4 in n≥ 4 variables. We identify the two minimal obstructions, at (n,d)=(4,4) and (n,d)=(5,3), and propagate them throughout the failure region by degree and variable aggregation.
ER -