Counting Dope Matrices

Abstract

For a polynomial $P$ of degree $n$ and an $m$-tuple $\Lambda =({\lambda }_1,\dots ,{\lambda }_m)$ of distinct complex numbers, the dope matrix of $P$ with respect to $\Lambda$ is $D_P(\Lambda )=({\delta }_{ij}{)}_{i\in [1,m],j\in [0,n]}$, where ${\delta }_{ij}=1$ if $P^{(j)}({\lambda }_i)=0$, and ${\delta }_{ij}=0$ otherwise. Our first result is a combinatorial characterization of the $2$-row dope matrices (for all pairs $\Lambda$); 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 $\Lambda$ is maximized when $\Lambda$ is generic. Finally, we resolve an ``extension'' problem of Nathanson and present several open problems.