AI 中文总结
该数学注记证明半代数集上的有界比率集合为有理多面体凸锥，给出洛伦兹多项式有界比率相关猜想的反例，关联7顶点割锥的非超度量团-网面。
AI 中文摘要
我们证明，半代数集$X\subset\R^n_{>0}$上的有界比率集合$\BR(X)$是在热带簇$\trop(X)$上非负的线性形式构成的凸锥，特别地，它是有理多面体凸锥。当$X$是具有固定M-凸支撑的洛伦兹多项式集合时，它是M-凸函数集合的对偶。我们给出了Huang--Huh--Soskin--Wang关于洛伦兹多项式有界比率猜想的明确反例，该反例中的有界比率对应7顶点割锥上的非超度量团-网面$\mathrm{CW}^1_7(1,1,1,1,1,-1,-1)$。
英文摘要
We prove that the set of bounded ratios $\BR(X)$ on a semialgebraic set $X\subset\R^n_{>0}$ is the convex cone of linear forms that are nonnegative on the tropicalization $\trop(X)$. In particular, it is a rational polyhedral convex cone. For $X$ the set of Lorentzian polynomials with fixed M-convex support, it is the dual to the set of M-convex functions. We record an explicit counterexample to a conjecture of Huang--Huh--Soskin--Wang on the bounded ratios on Lorentzian polynomials. The bounded ratio in the counterexample corresponds to the non-hypermetric clique-web facet $\mathrm{CW}^1_7(1,1,1,1,1,-1,-1)$ of the cut cone on seven vertices.
