AI 中文总结
本文研究洛伦兹多项式与三角超域上表示的定量关系，证明对于每个matroid M，存在q>0使得P L_M位于两个薄施鲁德细胞之间，并确定q(n)的渐进行为。
AI 中文摘要
Brändén和Huh证明洛伦兹多项式统一了组合学中的Hodge-Riemann关系：它们的支撑是M-凸的，并且每一个M-凸集都支持一个洛伦兹多项式。Baker、Huh、Kummer和Lorscheid后来证明，对于每一个q>0，洛伦兹多项式支撑J的项目化空间P L_J与弱表示J的薄施鲁德细胞Gr^w_J(T_q)是同胚的。我们研究洛伦兹多项式与三角超域上表示之间的定量关系。对于每一个matroid M，我们证明存在一个依赖于M的q>0，使得Gr^w_M(T_q)⊆P L_M⊆Gr^w_M(T_2)。因此，P L_M位于两个薄施鲁德细胞之间，每个都同胚于它。更一般地，对于每一个M-凸集J，存在一个依赖于J的q>0，使得N Gr^w_J(T_q)⊆P L_J⊆N Gr^w_J(T_2)，其中N表示归一化。我们还研究q(M)：=sup{q>0:Gr^w_M(T_q)⊆P L_M}。对于q(n)：=q(U_{2,n})，我们证明q(4)=2且q(5)=log_2 3，有匹配的上界和下界，顺序为1/n；因此q(n)=Θ(1/n)，所以尤其不存在q(n)的普遍正下界。
英文摘要
Brändén and Huh showed that Lorentzian polynomials unify Hodge-Riemann relations in combinatorics: their supports are M-convex, and every M-convex set supports a Lorentzian polynomial. Baker, Huh, Kummer, and Lorscheid later proved that, for every $q>0$, the projectivized space $\mathbf{P}\operatorname{L}_J$ of Lorentzian polynomials with support $J$ is homeomorphic to the thin Schubert cell $\operatorname{Gr}^{\mathrm{w}}_J(\mathbb{T}_q)$ of weak representations of $J$ over the generalized triangular hyperfield $\mathbb{T}_q$. We study the quantitative relation between Lorentzian polynomials and representations over triangular hyperfields. For every matroid $M$, we prove that some $q>0$ depending on $M$ satisfies $\operatorname{Gr}^{\mathrm{w}}_M(\mathbb{T}_q)\subseteq\mathbf{P}\operatorname{L}_M\subseteq\operatorname{Gr}^{\mathrm{w}}_M(\mathbb{T}_2)$. Thus $\mathbf{P}\operatorname{L}_M$ lies between two thin Schubert cells, each homeomorphic to it. More generally, for every M-convex set $J$, some $q>0$ depending on $J$ satisfies $\operatorname{N}\operatorname{Gr}^{\mathrm{w}}_J(\mathbb{T}_q)\subseteq\mathbf{P}\operatorname{L}_J\subseteq\operatorname{N}\operatorname{Gr}^{\mathrm{w}}_J(\mathbb{T}_2)$, where $\operatorname{N}$ denotes normalization. We also study $q(M):=\sup\{q>0:\operatorname{Gr}^{\mathrm{w}}_M(\mathbb{T}_q)\subseteq\mathbf{P}\operatorname{L}_M\}$. For $q(n):=q(U_{2,n})$, we prove $q(4)=2$ and $q(5)=\log_2 3$, with matching upper and lower bounds of order $1/n$; hence $q(n)=Θ(1/n)$, so in particular no universal positive lower bound for $q(n)$ exists.
Comments88 pages. v2: Minor changes to the Appendix
