关于Dressian的胞腔数目与维数的界
Bounds on the number of cells and the dimension of the Dressian
AI总结：
该研究针对拟阵的Dressian多面体复形，给出了其胞腔数目和维数的上界公式，还推导了结合拟阵结构的精细界，且均匀拟阵下的上界与稀疏铺砌拟阵导出的下界量级相当。
摘要：
拟阵$M$的{\rm Dressian}（德雷斯拟阵赋值空间）是$M$的所有赋值构成的集合。该Dressian是多面体复形$\boldsymbol{\rm Dr}(M)$的支撑集，其开胞腔与$M$的拟阵多面体的拟阵细分一一对应。我们给出了$\boldsymbol{\rm Dr}(M)$的胞腔数目与维数的上界。对于元素个数为$n$、秩$r\boldsymbol{\rm geq}3$的拟阵$M$，我们证明了当$n\boldsymbol{\rm ightarrow}\boldsymbol{\rm infty}$时，$\boldsymbol{\rm ln}\boldsymbol{\rm \\\\#}\boldsymbol{\rm Dr}(M)\boldsymbol{\rm leq}\binom{n}{r}O\left(\frac{\ln(n)^2}{n}\right)$，且$\boldsymbol{\rm dim}\boldsymbol{\rm Dr}(M)\boldsymbol{\rm leq}\binom{n}{r}\frac{3}{n-r+3}$，同时还给出了结合这类$M$结构性质的更精细界。对于均匀拟阵$M=U(r,n)$，这些上界与由稀疏铺砌拟阵构造的赋值所导出的下界相当。
英文摘要：
The {\em Dressian} of a matroid $M$ is the set of all valuations of $M$. This Dressian is the support of a polyhedral complex $\mathcal{Dr}(M)$ whose open cells correspond 1-1 with matroid subdivisions of the matroid polytope of $M$. We present upper bounds on the number of cells and the dimension of $\mathcal{Dr}(M)$. For matroids $M$ of rank $r\geq 3$ on $n$ elements we show that $$ \ln\#\mathcal{Dr}(M)\leq {\binom{n}{r}} O\left(\frac{\ln(n)^2}{n}\right)\text{ as }n\rightarrow\infty,\qquad\text{and}\qquad\dim \mathcal{Dr}(M)\leq {\binom{n}{r}}\frac{3}{n-r+3},$$ as well as some more detailed bounds that incorporate structural properties of such $M$. For uniform matroids $M=U(r,n)$, these upper bounds are comparable to lower bounds derived from valuations that are constructed from sparse paving matroids.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
