双线性型子空间的秩与维数的关系
Connections between rank and dimension for subspaces of bilinear forms
摘要：
设$K$是一个域，$V$是$K$上维数为$n$的向量空间。设$M$是定义在$V\times V$上的双线性型的子空间.令$r$是在$M$的元素之间出现的不同非零秩的数量。我们的目的是在各种假设下，根据$r$和$n$得到$\dim M$的上界。作为我们证明的一个例子，我们提到以下内容。假设$m$是作为$M$的元素的秩出现的最大整数。如果有$m\leq \lceil n/2\rceil$和$|K|\geq m+1$，我们就有$\dim M\leq rn$。$r=1$的情况对应于一个常秩空间，并证明了当$M$是一个常秩$m$空间时，$\dim M\leq n$和$|K|\geq m+1$。本文证明了当$|K|\geq m+1$和$K$是有限的或$K$具有不同于2的特征且$M$由对称形式组成时，常秩$m$空间$M$的维数界成立.在一般情况下，我们证明了，如果$M$是一个常数秩$m$子空间和$|K|\geq m+1$，那么$\dim M\leq \max\,(n,2m-1)$。我们还提供了有限域上常秩子空间，特别是交替或对称双线性型的子空间的更详细的结果。
英文摘要：
Let $K$ be a field and let $V$ be a vector space of dimension $n$ over $K$. Let $M$ be a subspace of bilinear forms defined on $V\times V$. Let $r$ be the number of different non-zero ranks that occur among the elements of $M$. Our aim is to obtain an upper bound for $\dim M$ in terms of $r$ and $n$ under various hypotheses. As a sample of what we prove, we mention the following. Suppose that $m$ is the largest integer that occurs as the rank of an element of $M$. Then if $m\leq \lceil n/2\rceil$ and $|K|\geq m+1$, we have $\dim M\leq rn$. The case $r=1$ corresponds to a constant rank space and it is conjectured that $\dim M\leq n$ when $M$ is a constant rank $m$ space and $|K|\geq m+1$. We prove that the dimension bound for a constant rank $m$ space $M$ holds provided $|K|\geq m+1$ and either $K$ is finite or $K$ has characteristic different from 2 and $M$ consists of symmetric forms. In general, we show that if $M$ is a constant rank $m$ subspace and $|K|\geq m+1$, then $\dim M\leq \max\,(n,2m-1)$. We also provide more detailed results about constant rank subspaces over finite fields, especially subspaces of alternating or symmetric bilinear forms.
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。
