  An $m \times (n+1)$ multiplicity matrix is a matrix $M = ( \mu_{i,j} )$ 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 $\mu_{i,j} \geq 1$, then $j \leq n-1$ and
$\mu_{i,j+1} = \mu_{i,j}-1$, and (2) the $j$th column sum of $M$ satisfies the
inequality $\sum_{i=1}^{m} \mu_{i, j} \leq n-j$ 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 $\Lambda = ( \lambda_1,\ldots, \lambda_{m})$ be a sequence of
distinct elements of $K$. For $i \in \{1, 2, \ldots, m \}$ and $j \in
\{1,2,\ldots, n\}$, let $ \mu_{i,j}$ be the multiplicity of $\lambda_i$ as a
zero of the polynomial $f^{(j)}(x)$. The $m \times (n+1)$ matrix $M_f(\Lambda)
= ( \mu_{i,j} )$ is called the multiplicity matrix of the polynomial $f(x)$
with respect to $\Lambda$. 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.
