论文标题
计数涂料矩阵
Counting Dope Matrices
论文作者
论文摘要
对于多项式$ p $度$ n $和$ m $ -tuple $λ=（λ_1，\ dots，λ_m）$的$ p $，相对于$λ$的涂料矩阵是$λ$ as $ d_p（λ）=（Δ_{ij {ij {ij {ij {ij {ij {ij {ij} $） $δ_{ij} = 1 $如果$ p^{（j）}（λ_i）= 0 $，而$δ_{ij} = 0 $否则。我们的第一个结果是对$ 2 $ - 行涂料矩阵的组合表征（对于所有对$λ$）；使用此表征，我们解决了相关的枚举问题。我们还提供了$ m \ times（n+1）$涂料矩阵的上限，并且我们表明$ m \ times（n+1）$ dope矩阵的数量是固定$ m $ $ $ tuple $λ$当$λ$是通用时的最大化。最后，我们解决了Nathanson的``扩展''问题，并提出了几个开放问题。
For a polynomial $P$ of degree $n$ and an $m$-tuple $Λ=(λ_1,\dots,λ_m)$ of distinct complex numbers, the dope matrix of $P$ with respect to $Λ$ is $D_P(Λ)=(δ_{ij})_{i\in [1,m],j\in[0,n]}$, where $δ_{ij}=1$ if $P^{(j)}(λ_i)=0$, and $δ_{ij}=0$ otherwise. Our first result is a combinatorial characterization of the $2$-row dope matrices (for all pairs $Λ$); using this characterization, we solve the associated enumeration problem. We also give upper bounds on the number of $m\times(n+1)$ dope matrices, and we show that the number of $m \times (n+1)$ dope matrices for a fixed $m$-tuple $Λ$ is maximized when $Λ$ is generic. Finally, we resolve an ``extension'' problem of Nathanson and present several open problems.
