论文标题
线性操作员在有限字段上的分裂子空间
Splitting Subspaces of Linear Operators over Finite Fields
论文作者
论文摘要
令$ v $为有限字段$ \ mathbb {f} _q $和$ t $的矢量空间$ n $是$ v $的线性运算符。给定一个划分$ n $的整数$ m $，如果$ v = $ v = w \ oplus tw \ oplus \ cdots \ cdots \ oplus \ oplus t^{d-1} w $ n $ d $ d $ d = n $ d = n/m $ d = n/m $。令$σ（m，d; t）$表示$ m $二维$ t $ splitting子空间的数量。确定任意运算符$ t $的$σ（M，D; T; T）$是一个开放问题。我们证明，$σ（M，D; T）$仅取决于$ t $的相似性类型，并在$ t $ cyclic和nilpotent的特殊情况下给出明确的公式。用$σ_Q（m，d;τ）$表示$ \\ MATHBB {f} _Q $ -VECTOR-dimension $ MD $的线性类型$τ$的线性运算符的$ m $二维分割子空间的数量。对于$ m，d $和$τ$的固定值，我们表明$σ_q（m，d;τ）$是$ q $的多项式。
Let $V$ be a vector space of dimension $N$ over the finite field $\mathbb{F}_q$ and $T$ be a linear operator on $V$. Given an integer $m$ that divides $N$, an $m$-dimensional subspace $W$ of $V$ is $T$-splitting if $V=W\oplus TW\oplus \cdots \oplus T^{d-1}W$ where $d=N/m$. Let $σ(m,d;T)$ denote the number of $m$-dimensional $T$-splitting subspaces. Determining $σ(m,d;T)$ for an arbitrary operator $T$ is an open problem. We prove that $σ(m,d;T)$ depends only on the similarity class type of $T$ and give an explicit formula in the special case where $T$ is cyclic and nilpotent. Denote by $σ_q(m,d;τ)$ the number of $m$-dimensional splitting subspaces for a linear operator of similarity class type $τ$ over an $\\mathbb{F}_q$-vector space of dimension $md$. For fixed values of $m,d$ and $τ$, we show that $σ_q(m,d;τ)$ is a polynomial in $q$.
