发表机构
Oak Park High School; Department of Mathematics, UCLA（橡树公园高中; 加州大学洛杉矶分校数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究刻画了洛伦兹多项式的有界比率锥，确定了三元洛伦兹三次多项式的最优界常数，明确了可热带化计算有界比率锥的(n,k)对，并得出任意次数三元情形下锥的生成元为三角比率的结论。
AI 中文摘要
我们通过“有界比率”的概念研究洛伦兹多项式系数之间的乘性不等式。主要结果完全刻画了k个变量的n次洛伦兹多项式的有界比率锥，该对偶刻画基于M-凸函数模仿射函数的等价类表达。对于三元洛伦兹三次多项式，我们确定了每个有界比率的最优界常数；还刻画了(n,k)对，其中有界比率锥可通过热带化k个变量的n个非负线性形式的乘积来计算。此外，证明了任意次数n的三元情形下，有界比率锥具有相当简单的生成元，即三角比率。
英文摘要
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.
CommentsWe expanded concurrent work paragraph
