Counting Dope Matrices

TL;DR

This paper introduces a combinatorial framework for analyzing dope matrices associated with polynomials and complex points, providing enumeration, bounds, and optimal configurations, and addressing an extension problem with open questions.

Contribution

It offers a combinatorial characterization of 2-row dope matrices, solves their enumeration, and establishes bounds and optimality conditions for the number of dope matrices.

Findings

Characterization of 2-row dope matrices for all point sets

Enumeration of dope matrices based on this characterization

Maximum number of dope matrices occurs at generic point configurations

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.