论文标题
毒矩阵的通用分类和渐近枚举
Generic Classification and Asymptotic Enumeration of Dope Matrices
论文作者
论文摘要
对于一个复杂的多项式$ p $度$ n $和一个$ m $ $ $ $ $λ=（λ_1，\ ldots，λ_m）$，涂料矩阵$ d_p（λ）$定义为$ m \ times（n+1）$（n+1）$（c） $ p^{（j）}（λ_i）= 0 $和$ c_ {ij} = 0 $否则。当$λ$的条目在代数上独立时，我们对浓汤矩阵进行了分类，解决了Alon，Kravitz和O'Bryant的猜想。我们还提供$ m \ times（n+1）$涂料矩阵的渐近上限和下限。对于$ M $大于$ n $，这些界限给出了$ m \ times（n+1）$涂料矩阵的对数的渐近估计。
For a complex polynomial $P$ of degree $n$ and an $m$-tuple of distinct complex numbers $Λ=(λ_1,\ldots,λ_m)$, the dope matrix $D_P(Λ)$ is defined as the $m \times (n+1)$ matrix $(c)_{ij}$ with $c_{ij} =1$ if $P^{(j)}(λ_i)=0$ and $c_{ij}=0$ otherwise. We classify the set of dope matrices when the entries of $Λ$ are algebraically independent, resolving a conjecture of Alon, Kravitz, and O'Bryant. We also provide asymptotic upper and lower bounds on the total number of $m \times (n+1)$ dope matrices. For $m$ much smaller than $n$, these bounds give an asymptotic estimate of the logarithm of the number of $m \times (n+1)$ dope matrices.
