Bounded ratios for Lorentzian polynomials

Abstract

We study multiplicative inequalities among the coefficients of Lorentzian polynomials through the notion of bounded ratios. Our main structural result completely characterizes the cone of bounded ratios for Lorentzian polynomials of degree $n$ in $k$ variables. We show that the dual of the cone of bounded ratios is generated by equivalence classes of M-convex functions modulo affine functions. For ternary Lorentzian forms of arbitrary degree $n\ge3$, we show that the cone of bounded ratios is generated by triangular ratios and determine the optimal bounding constant of every bounded ratio. Furthermore, we characterize the pairs $(n,k)$ for which the cone of bounded ratios can be computed by tropicalizing products of $n$ linear forms in $k$ variables with nonnegative coefficients.