多项式的几何组合学I：单一多项式的情形
Geometric Combinatorics of Polynomials I: The Case of a Single Polynomial
摘要：
有许多不同的代数，几何和组合的对象，可以附加到一个复杂的多项式与不同的根。在这篇文章中，我们介绍了一个新的对象，编码许多现有的对象，以前出现在文献中。具体地说，对于每个具有$d$不同根且次数至少为2的复多项式$p$，我们产生一个规范的紧凑平面2-复形，它是瓦片相图的紧凑度量版本。它有一个局部CAT（0）度量，该度量与由$p$的临界点索引的有限内点集局部欧氏距离，并且它的每个2-胞是一个度量矩形。从这个平面矩形2-复形中，我们可以使用度量图（称为度量仙人掌和度量榕树）来读出几段组合数据：分划格中的链、d-圈的循环因式分解、实非交叉分划（也称为本原d-较多）以及多项式的单值置换。这篇文章是一个系列的第一篇。
英文摘要：
There are many different algebraic, geometric and combinatorial objects that one can attach to a complex polynomial with distinct roots. In this article we introduce a new object that encodes many of the existing objects that have previously appeared in the literature. Concretely, for every complex polynomial $p$ with $d$ distinct roots and degree at least 2, we produce a canonical compact planar 2-complex that is a compact metric version of a tiled phase diagram. It has a locally CAT(0) metric that is locally Euclidean away from a finite set of interior points indexed by the critical points of $p$, and each of its 2-cells is a metric rectangle. From this planar rectangular 2-complex one can use metric graphs known as metric cacti and metric banyans to read off several pieces of combinatorial data: a chain in the partition lattice, a cyclic factorization of a d-cycle, a real noncrossing partition (also known as a primitive d-major), and the monodromy permutations for the polynomial. This article is the first in a series.
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。
