有向树的路径多项式偏序集
Posets of trek polynomials for directed trees
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中文总结 AI 辅助
本文刻画了枚举有向树路径子图的生成多项式对应的偏序集，借此解决了图模型程序中结构可识别性问题的一个实例。
中文摘要 AI 辅助
当簇$V_\varphi$等于多项式映射$\varphi$的像，且其坐标函数是组合生成多项式（即枚举组合对象的多项式）时，$V_\varphi$的几何性质反映了这些生成多项式满足的恒等式。组合数学与代数几何之间由此产生的相互作用可用于解答关于$V_\varphi$的问题。近期一项技术提出，通过由定义$\varphi$的多项式的系数向量构成的偏序集（poset）$P_\varphi$来实现这一目标。本文对当定义$\varphi$的生成多项式枚举称为路径（treks）的有向树子图时的偏序集$P_\varphi$进行了刻画。该刻画被用于计算$V_\varphi$的线性张成，证明其为环面簇，并推导其零化理想的一组基。还证明了有向树的路径多项式偏序集是所谓的$\pi$-系统，当且仅当该树满足由Stanley的P-分拆刻画的性质。作为额外结论，两个不同有向树对应的簇相交于严格低维的簇，这解决了统计学科图模型程序中结构可识别性问题的一个实例。
英文摘要
When a variety $V_φ$ equals the image of a polynomial map $φ$ whose coordinate functions are combinatorial generating polynomials (i.e.~polynomials enumerating combinatorial objects), the geometry of $V_φ$ reflects identities satisfied by the generating polynomials. The resulting interplay between combinatorics and algebraic geometry can be used to answer questions about $V_φ$. A recent technique proposes to do so using a partially ordered set (poset) $P_φ$ defined via the coefficient vectors of the polynomials defining $φ$. This paper characterizes the poset $P_φ$ when the generating polynomials defining $φ$ enumerate subgraphs of a directed tree known as treks. The characterization is used to compute the linear span of $V_φ$, prove it is toric and deduce a basis for its vanishing ideal. It is also shown that this poset of trek polynomials for a directed tree is a so-called $π$-system if and only if the tree satisfies a property characterized via Stanley's P-partitions. As an additional consequence, it is shown that the varieties for two distinct directed trees intersect in a strictly lower-dimensional variety. This solves an instance of the structural identifiability problem in the graphical models program from statistics.
