Counting Dope Matrices

Abstract

For a polynomial $P$ of degree $n$ and an $m$-tuple $Λ=(λ_1,\dots,λ_m)$ of distinct complex numbers, the dope matrix of $P$ with respect to $Λ$ is $D_P(Λ)=(δ_{ij})_{i\in [1,m],j\in[0,n]}$, where $δ_{ij}=1$ if $P^{(j)}(λ_i)=0$, and $δ_{ij}=0$ otherwise. Our first result is a combinatorial characterization of the $2$-row dope matrices (for all pairs $Λ$); using this characterization, we solve the associated enumeration problem. We also give upper bounds on the number of $m\times(n+1)$ dope matrices, and we show that the number of $m \times (n+1)$ dope matrices for a fixed $m$-tuple $Λ$ is maximized when $Λ$ is generic. Finally, we resolve an ``extension'' problem of Nathanson and present several open problems.