论文标题
多项式和多重矩阵的零的相互作用
Interactions of zeros of of polynomials and multiplicity matrices
论文作者
论文摘要
$ m \ times（n+1）$多样性矩阵是矩阵$ m =（μ_{i，j}）$，由$ i \ in \ in \ in \ {1，\ 2，\ ldots，m \} $ ed列出行，并由$ j \ in \ in \ in \ nonnon nonnon nonnon nonnon nonny nonnon nonnon nonnon nonny nonnon nonny nonno满足以下两个属性的整数：（1）如果$μ_{i，j} \ geq 1 $，然后$ j \ leq n-1 $和$μ__{i，j+1} =μ_{i，j} -1 $，（2）$ j $ j $ j $ th列的$ m $满足$ sum__}的emem_} {MO} {MO} μ_{i，j} \ leq n-j $用于所有$ j $。 令$ k $为特征0的字段，让$ f（x）$为$ k $的系数$ n $的多项式。令$ f^{（j）}（x）$为$ f（x）$的$ j $ th衍生物。令$λ=（λ_1，\ ldots，λ_{m}）$为$ k $的一系列不同元素。对于$ i \ in \ {1，2，\ ldots，m \} $和$ j \ in \ {1,2，\ ldots，n \} $，让$μ__{i，j} $是$λ_i$的多重性，为polynomial $ f^{（j）的零。 $ m \ times（n+1）$矩阵$ m_f（λ）=（μ_{i，j}）$称为$λ$的多项式$ f（x）$的多重矩阵。一个开放的问题是对$ k [x] $中多项式的多重矩阵进行分类，并构造不是多项式多样性矩阵的多重矩阵。
An $m \times (n+1)$ multiplicity matrix is a matrix $M = ( μ_{i,j} )$ with rows enumerated by $i \in \{ 1,\ 2, \ldots, m \}$ and columns enumerated by $j \in \{ 0,1,\ldots, n \}$ whose coordinates are nonnegative integers satisfying the following two properties: (1) If $μ_{i,j} \geq 1$, then $j \leq n-1$ and $μ_{i,j+1} = μ_{i,j}-1$, and (2) the $j$th column sum of $M$ satisfies the inequality $\sum_{i=1}^{m} μ_{i, j} \leq n-j$ for all $j$. Let $K$ be a field of characteristic 0 and let $f(x)$ be a polynomial of degree $n$ with coefficients in $K$. Let $f^{(j)}(x)$ be the $j$th derivative of $f(x)$. Let $Λ= ( λ_1,\ldots, λ_{m})$ be a sequence of distinct elements of $K$. For $i \in \{1, 2, \ldots, m \}$ and $j \in \{1,2,\ldots, n\}$, let $ μ_{i,j}$ be the multiplicity of $λ_i$ as a zero of the polynomial $f^{(j)}(x)$. The $m \times (n+1)$ matrix $M_f(Λ) = ( μ_{i,j} )$ is called the multiplicity matrix of the polynomial $f(x)$ with respect to $Λ$. An open problem is to classify the multiplicity matrices that are multiplicity matrices of polynomials in $K[x]$ and to construct multiplicity matrices that are not multiplicity matrices of polynomials.
