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关于通过一点将某种类型秩3的均匀定向矩阵体约化的研究

On the Reduciblity of a Certain Type of Rank 3 Uniform Oriented Matroid by a Point

Harish-Chandra Research Institute（哈里什-昌德拉研究所）

C P Anil Kumar

备注：
27 pages, 5 figures
发表信息：
Proceedins Mathematical Sciences: The Indian Academy of Sciences, Vol. 136, Art. ID. 11, 2026, https://www.ias.ac.in/article/fulltext/pmsc/136/0011
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AI总结：

本文研究了凸n边形内部区域与对称群S_n中特定类型的n循环的关系，发现并非所有区域标签的循环都出现在每个凸n边形中，进而定义了确定性和不定性循环，并探讨了其在均匀秩3凸定向矩阵体的一点扩展中的可约性。

摘要：

对于正整数n≥3，凸n边形的边和对角线将内部分成有限多个区域。本文通过研究平面中的点排列，将每个区域与对称群S_n中的特定类型的n循环关联起来。研究发现，存在指数级数量的此类循环，导致并非每个区域标签的循环都出现在每个凸n边形中。事实上，大多数循环在任何给定的凸n边形中都不出现。本文的主要定理通过组合方法，刻画了那些对应的区域出现在每个凸n边形中的循环（定义为确定性循环）以及那些对应的区域不出现在每个凸n边形中的循环（定义为非确定性循环）。作为结果，本文还刻画了那些一点扩展为均匀秩3凸定向矩阵体的扩展，当该点位于凸包内时，扩展是可约的。

英文摘要：

For a positive integer $n\geq 3$, the sides and diagonals of a convex $n$-gon divide the interior of the convex $n$-gon into finitely (polynomial in $n$) many regions bounded by them. In this article, we associate to every region a unique $n$-cycle in the symmetric group $S_n$ of a certain type (defined as $2$-standard consecutive cycle) by studying point arrangements in the plane. Then we find that there are more (exponential in $n$) number of such cycles leading to the conclusion that not every region labelled by a cycle appears in every convex $n$-gon. In fact most of them do not occur in any given single convex $n$-gon. Later in the main theorem of this article we characterize combinatorially those cycles (defined as definite cycles) whose corresponding regions occur in every convex $n$-gon and those cycles (defined as indefinite cycles) whose corresponding regions do not occur in every convex $n$-gon. As a consequence we characterize those one point extensions of a uniform rank $3$ convex oriented matroid for which the one point extension is reducible by the, one point, when it lies inside the convex hull.

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。

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