Generic classification and asymptotic enumeration of dope matrices

Abstract

AbstractFor a complex polynomial of degree and an ‐tuple of distinct complex numbers , the dope matrix is defined as the matrix with if and otherwise. We classify the set of dope matrices when the entries of 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 dope matrices. For much smaller than , these bounds give an asymptotic estimate of the logarithm of the number of dope matrices.