Generic Classification and Asymptotic Enumeration of Dope Matrices

TL;DR

This paper classifies dope matrices associated with complex polynomials and algebraically independent points, resolving a conjecture and providing asymptotic bounds on their total number, advancing understanding in polynomial zero patterns.

Contribution

It classifies dope matrices for algebraically independent points and establishes asymptotic bounds on their count, resolving a prior conjecture.

Findings

Complete classification of dope matrices for algebraically independent points.

Asymptotic bounds on the number of dope matrices.

Resolution of a conjecture by Alon, Kravitz, and O'Bryant.

Abstract

For a complex polynomial $P$ of degree $n$ and an $m$-tuple of distinct complex numbers $\Lambda=(\lambda_1,\ldots,\lambda_m)$, the dope matrix $D_P(\Lambda)$ 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 $\Lambda$ 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.