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Generic Classification and Asymptotic Enumeration of Dope Matrices

Ankit Bisain

TL;DR

The set of dope matrices when the entries of are algebraically independent is classified, resolving a conjecture of Alon, Kravitz, and O'Bryant.

Abstract

For a complex polynomial $P$ of degree $n$ and an $m$-tuple of distinct complex numbers $Λ=(λ_1,\ldots,λ_m)$, the dope matrix $D_P(Λ)$ is defined as the $m \times (n+1)$ matrix $(c)_{ij}$ with $c_{ij} =1$ if $P^{(j)}(λ_i)=0$ and $c_{ij}=0$ otherwise. We classify the set of dope matrices when the entries of $Λ$ 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.

Generic Classification and Asymptotic Enumeration of Dope Matrices

TL;DR

The set of dope matrices when the entries of are algebraically independent is classified, resolving a conjecture of Alon, Kravitz, and O'Bryant.

Abstract

For 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.
Paper Structure (14 sections, 26 theorems, 65 equations)

This paper contains 14 sections, 26 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Generic Dope Matrices
  4. Reduction to the Key Lemma
  5. Demonstration of Proof Technique
  6. Derivatives as Linear Combinations
  7. Combining Roots
  8. Enumeration
  9. $\mathcal{D}_n^m$ for Small $m$
  10. Upper Bound
  11. Lower Bound
  12. $\mathcal{D}_n^m$ for Large $m$
  13. Future Work
  14. Acknowledgments

Key Result

Theorem 1.1

For all positive integers $m,n$, the set $\mathcal{D}_n^{\mathop{\mathrm{gen}}\nolimits(m)}$ consists of exactly the $m \times (n+1)$ matrices with $\{0,1\}$ entries such that for all $k \in [0,n]$, there are at most $k$ nonzero entries in the last $k+1$ columns.

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 1.5
  • proof
  • Lemma 1.6
  • Theorem 1.7
  • proof
  • Theorem 2.1
  • ...and 39 more