Interactions of zeros of of polynomials and multiplicity matrices

TL;DR

This paper studies the structure of multiplicity matrices derived from polynomial zeros and derivatives, aiming to classify which matrices correspond to actual polynomials and which do not.

Contribution

It introduces a formal framework for multiplicity matrices and explores conditions for their realizability as polynomial zero multiplicity patterns.

Findings

Defined properties of multiplicity matrices and their constraints.

Established criteria for when a matrix can be realized by a polynomial.

Identified open problems in classifying and constructing multiplicity matrices.

Abstract

An $m \times (n+1)$ multiplicity matrix is a matrix $M = ( \mu_{i,j} )$ with rows enumerated by $i \in \{ 1,\ 2, \ldots, m \}$ 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.