Bounded ratios for Lorentzian polynomials

Abstract: We study multiplicative inequalities among the coefficients of Lorentzian polynomials through the notion of \emph{bounded ratios}. Our main result completely characterizes the cone of bounded ratios for Lorentzian polynomials of degree $n$ in $k$ variables. This dual characterization is expressed in terms of equivalence classes of M-convex functions modulo affine functions. For ternary Lorentzian cubics, we determine the optimal bounding constant of every bounded ratio. We also characterize the pairs $(n,k)$ for which the bounded-ratio cone can be computed by tropicalizing products of $n$ nonnegative linear forms in $k$ variables. Furthermore, show that in the ternary case of any degree n the cone of bounded ratios has rather simple generators which are triangular ratios.