Interactions of zeros of polynomials and multiplicity matrices

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