论文标题
矩形I的基础：没有大均匀未成年人的矩形
Foundations of matroids I: Matroids without large uniform minors
论文作者
论文摘要
Matroid的基础是一个规范的代数不变式，将矩阵的表示形式分类为重新等价。基金会的基础是牧场，同时概括了部分场和超场。 通过Tutte，Dress-Wenzel和Gelfand-Rybnikov-Stone，使用深度结果，我们在发电机和关系方面为Matroid的基础提供了演讲。发电机是一定的“交叉比例”，可以在投影线上概括四个点的交叉比例，并且在某些低级别构型中的交叉比例之间的关系依赖关系在投射几何形状中产生。 尽管基础的呈现对于所有矩形都是有效的，但在没有大统一未成年人的情况下，最简单的应用是最简单的，即没有较小对应于一条线上的五个点或其双重配置的矩形。对于此类矩阵，我们获得了所有可能的基础的完整分类。然后，我们提供此分类定理的许多应用程序，例如： - 我们证明了Lee和Scobee定理的以下加强：具有大型统一未成年人的矩形的每个方向都来自二元代表，这是独特的重新定位。 - 对于具有大型统一未成年人的矩阵$ m $，我们建立了以下2017年Ardila-rincón-Williams定理的加强：如果$ m $呈阳性定向，则$ m $在每个领域都可以在每个领域中具有至少三个元素的代表。 - 如果两个矩形属于同一表示类别，则它们属于同一牧场。我们证明，对于没有大统一的未成年人的代表类别的代表类别，恰好有12个可能性，其中三个在任何领域都无法代表。
The foundation of a matroid is a canonical algebraic invariant which classifies representations of the matroid up to rescaling equivalence. Foundations of matroids are pastures, a simultaneous generalization of partial fields and hyperfields. Using deep results due to Tutte, Dress-Wenzel, and Gelfand-Rybnikov-Stone, we give a presentation for the foundation of a matroid in terms of generators and relations. The generators are certain "cross-ratios" generalizing the cross-ratio of four points on a projective line, and the relations encode dependencies between cross-ratios in certain low-rank configurations arising in projective geometry. Although the presentation of the foundation is valid for all matroids, it is simplest to apply in the case of matroids without large uniform minors, i.e. matroids having no minor corresponding to five points on a line or its dual configuration. For such matroids, we obtain a complete classification of all possible foundations. We then give a number of applications of this classification theorem, for example: - We prove the following strengthening of a theorem of Lee and Scobee: every orientation of a matroid without large uniform minors comes from a dyadic representation, which is unique up to rescaling. - For a matroid $M$ without large uniform minors, we establish the following strengthening of a 2017 theorem of Ardila-Rincón-Williams: if $M$ is positively oriented then $M$ is representable over every field with at least three elements. - Two matroids are said to belong to the same representation class if they are representable over precisely the same pastures. We prove that there are precisely 12 possibilities for the representation class of a matroid without large uniform minors, exactly three of which are not representable over any field.
