Generic Classification and Asymptotic Enumeration of Dope Matrices

For a complex polynomial $P$ of degree $n$ and an $m$-tuple of distinct complex numbers $\mathrm{\Lambda }=({\lambda }_1,\dots ,{\lambda }_m)$, the dope matrix $D_P(\mathrm{\Lambda })$ is defined as the $m\times (n+1)$ matrix $(c{)}_{ij}$ with $c_{ij}=1$ if $P^{(j)}({\lambda }_i)=0$ and $c_{ij}=0$ otherwise. We classify the set of dope matrices when the entries of $\mathrm{\Lambda }$ are algebraically independent, resolving a conjecture of Alon, Kravitz, and O’Bryant. We also provide asymptotic upper and lower bounds on the total number of $m\times (n+1)$ dope matrices. For $m$ much smaller than $n$, these bounds give an asymptotic estimate of the logarithm of the number of $m\times (n+1)$ dope matrices.