Multiplicity matrices of polynomials

Let $f(x)$ be a polynomial of degree $n$ and let $f^{(j)}(x)$ be the $j$th derivative of $f(x)$.

Let $\Lambda = (\lambda_1, \ldots , \lambda_m)$ be a strictly increasing sequence of real numbers.

For $i\in \{1, \ldots , m\}$ and $j\in \{0, 1, \ldots , n\}$, let $\mu_{i,j}$ be the multiplicity of $\lambda_i$ as a root of the polynomial $f^{(j)}(x)$. The multiplicity matrix of $f$ with respect to $\lambda_1, \ldots , \lambda_m$ is the $m\times (n + 1)$ matrix $M_f (\Lambda) = (\mu_{i,j})$ .

The problem is to describe the matrices are multiplicity matrices of polynomials.

Mel Nathanson