向量空间中Manickam-Miklós-Singhi猜想的注记
A Note on the Manickam-Miklós-Singhi Conjecture for Vector Spaces
摘要：
设$V$是有限域$\mathbb{F}_q$上的$n$维向量空间.在$V$的$1$维向量空间上定义一个实值权重函数，使得所有权重之和为零。设子空间$S$的权重是包含在$S$中的$1$维子空间的权重之和。Manickam和Singhi在1988年证明了，如果$n \geq 4k$，那么具有非负权的$k$维子空间的个数至少等于固定$1$维子空间上的$k$维子空间的个数.最近Chowdhury，Huang，Sarkis，Shahriari和Sudakov证明了Manickam和Singhi关于$n \geq 3k$的猜想.我们修改了Chowdhury等人所使用的技术，证明了$n \geq 2k$的猜想，如果$q$是大的。此外，如果等式成立且$n \geq 2k+1$成立，则具有非负权的$k$维子空间的集合是固定$1$维子空间上的所有$k$维子空间的集合.
英文摘要：
Let $V$ be an $n$-dimensional vector space over a finite field $\mathbb{F}_q$. Define a real-valued weight function on the $1$-dimensional vector spaces of $V$ such that the sum of all weights is zero. Let the weight of a subspace $S$ be the sum of the weights of the $1$-dimensional subspaces contained in $S$. In 1988 Manickam and Singhi conjectured that if $n \geq 4k$, then the number of $k$-dimensional subspaces with nonnegative weight is at least the number of $k$-dimensional subspaces on a fixed $1$-dimensional subspace. Recently, Chowdhury, Huang, Sarkis, Shahriari, and Sudakov proved the conjecture of Manickam and Singhi for $n \geq 3k$. We modify the technique used by Chowdhury et al. to prove the conjecture for $n \geq 2k$ if $q$ is large. Furthermore, if equality holds and $n \geq 2k+1$, then the set of $k$-dimensional subspaces with nonnegative weight is the set of all $k$-dimensional subspaces on a fixed $1$-dimensional subspace.
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