Interactions of zeros of polynomials and multiplicity matrices

Abstract

An m×(n+1) multiplicity matrix is a matrix M=(μi,j) with rows enumerated by i∈{1,2,…,m} and columns enumerated by j∈{0,1,…,n} whose coordinates are nonnegative integers satisfying the following two properties: (1) If μi,j≥1, then j≤n−1 and μi,j+1=μi,j−1, and (2) colsumj(M)=∑i=1mμi,j≤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 jth derivative of f(x). Let Λ=(λ1,…,λm) be a sequence of distinct elements of K. For i∈{1,2,…,m} and j∈{1,2,…,n}, let μi,j be the multiplicity of λi as a zero of the polynomial f(j)(x). The m×(n+1) matrix Mf(Λ)=(μi,j) is called the multiplicity matrix of the polynomialf(x)with respect to Λ. Conditions for a multiplicity matrix to be the multiplicity matrix of a polynomial are established, and examples are constructed of multiplicity matrices that are not multiplicity matrices of polynomials. 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.