论文标题
$ Q $ -MATROID的Projectivization Matroid
The Projectivization Matroid of a $q$-Matroid
论文作者
论文摘要
在本文中，我们研究了$ q $ -MATROID及其相关的Matroid（称为Projectivization Matroid）之间的关系。后者是通过将$ Q $ -MATROID的地面空间进行预测而产生的，并将投影空间视为相关的Matroid的地面集，其定义了与$ Q $ -Matroid兼容的等级函数。我们表明，该项目映射是从$ Q $ -MATROID类别到Matroid类别的函数，该类别可以证明有关$ Q $ -MATROIDS的地图的新结果。我们此外还显示了$ q $ -Matroid的特征多项式等于Projectivization Matroid的特征多项式。我们使用这种关系来建立一个递归公式，以根据其未成年人的特征多项式来为$ q $ - amatroid的特征多项式建立递归公式。最后，我们使用projectivization Matroid来证明关键定理的$ q $ - nalogue，以$ \ mathbb {f} _ {q^m} $ - 线性等级度量代码和$ q $ -MATROIDS。
In this paper, we investigate the relation between a $q$-matroid and its associated matroid called the projectivization matroid. The latter arises by projectivizing the groundspace of the $q$-matroid and considering the projective space as the groundset of the associated matroid on which is defined a rank function compatible with that of the $q$-matroid. We show that the projectivization map is a functor from categories of $q$-matroids to categories of matroids, which allows to prove new results about maps of $q$-matroids. We furthermore show the characteristic polynomial of a $q$-matroid is equal to that of the projectivization matroid. We use this relation to establish a recursive formula for the characteristic polynomial of a $q$-matroid in terms of the characteristic polynomial of its minors. Finally we use the projectivization matroid to prove a $q$-analogue of the critical theorem in terms of $\mathbb{F}_{q^m}$-linear rank metric codes and $q$-matroids.
