论文标题
Rota的基础猜想对向量空间的随机基础存在
Rota's basis conjecture holds for random bases of vector spaces
论文作者
论文摘要
In 1989, Rota conjectured that, given $n$ bases $B_1,\dots,B_n$ of the vector space $\mathbb{F}^n$ over some field $\mathbb{F}$, one can always decompose the multi-set $B_1\cup \dots \cup B_n$ into transversal bases.尽管引起了很多关注，但这种猜想仍然敞开着。在本文中，我们考虑了随机基础的设置$ b_1，\ dots，b_n $。更具体地说，我们的第一个结果表明，如果基部$ b_1，\ dots，b_n $，Rota的基础猜想与$ 1-o（1）$ as $ 1-o（1）$ as $ n \ to \ infty $，则在所有$ \ alsbb {f} _q^n $的所有基础上随机独立地选择了$ \ Q^n $的所有基础，并结果是$ \ q^_q^n $。对于无限字段$ \ mathbb {f} $）。换句话说，对于几乎所有基础选择$ b_1，\ dots，b_n \ subseteq \ mathbb {f} _q^n $的选择都是正确的。我们的第二个更通用的结果涉及随机基础$ b_1，\ dots，b_n \ subseteq s^n $，对于某些给定有限的子集$ s \ subseteq \ subseteq \ mathbb {f} $（换句话说，bases $ b_1，\ dots $ b_1，\ dots，b_n $所有载体中所有载体都有$ s $ s $ $）。我们表明，当选择$ b_1，\ dots，b_n \ subseteq s^n $在所有基部的$ s^n $的子集中随机均匀地均匀地均匀地均匀地统一时，rota的基础猜想具有$ 1-o（1-o（1）$ as $ n \ to \ infty $。
In 1989, Rota conjectured that, given $n$ bases $B_1,\dots,B_n$ of the vector space $\mathbb{F}^n$ over some field $\mathbb{F}$, one can always decompose the multi-set $B_1\cup \dots \cup B_n$ into transversal bases. This conjecture remains wide open despite of a lot of attention. In this paper, we consider the setting of random bases $B_1,\dots,B_n$. More specifically, our first result shows that Rota's basis conjecture holds with probability $1-o(1)$ as $n\to \infty$ if the bases $B_1,\dots,B_n$ are chosen independently uniformly at random among all bases of $\mathbb{F}_q^n$ for some finite field $\mathbb{F}_q$ (the analogous result is trivially true for an infinite field $\mathbb{F}$). In other words, the conjecture is true for almost all choices of bases $B_1,\dots,B_n\subseteq \mathbb{F}_q^n$. Our second, more general, result concerns random bases $B_1,\dots,B_n\subseteq S^n$ for some given finite subset $S\subseteq \mathbb{F}$ (in other words, bases $B_1,\dots,B_n$ where all vectors have entries in $S$). We show that when choosing bases $B_1,\dots,B_n\subseteq S^n$ independently uniformly at random among all bases that are subsets of $S^n$, then again Rota's basis conjecture holds with probability $1-o(1)$ as $n\to \infty$.
