Interactions of zeros of of polynomials and multiplicity matrices

Abstract

An $m\times (n+1)$ multiplicity matrix is a matrix $M=({\mu }_{i,j})$ with rows enumerated by $i\in \{1,2,\dots ,m\}$ and columns enumerated by $j\in \{0,1,\dots ,n\}$ whose coordinates are nonnegative integers satisfying the following two properties: (1) If ${\mu }_{i,j}\ge 1$, then $j\le n-1$ and ${\mu }_{i,j+1}={\mu }_{i,j}-1$, and (2) the $j$th column sum of $M$ satisfies the inequality ${\sum }_{i=1}^m{\mu }_{i,j}\le n-j$ 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 $\Lambda =({\lambda }_1,\dots ,{\lambda }_m)$ be a sequence of distinct elements of $K$. For $i\in \{1,2,\dots ,m\}$ and $j\in \{1,2,\dots ,n\}$, 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.

arXiv comment17 pages; minor changes