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 $S\subseteq {\Delta }_n^d$. For $d\ge 2$, this quadratic local-to-global principle holds universally if and only if \[ n\leq 3,\qquad d=2,\qquad\text{or}\qquad (n,d)=(4,3). \] In every remaining case, the principle fails already for Lorentzian polynomials with full support: for cubics in $n\ge 5$ variables and for polynomials of degree $d\ge 4$ in $n\ge 4$ 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.

arXiv comment25 pages, we reconstructed and simplified the proof of the counterexamples and propagations, added the declaration of AI usage, added the citation of concurrent previous work as [4] and did other minor revisions