Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Statistics of ranks, determinants and characteristic polynomials of rational matrices

Muhammad Afifurrahman, Vivian Kuperberg, Alina Ostafe, Igor E. Shparlinski

Abstract

We consider the set of $n\times n$ matrices with rational entries having numerator and denominator of size at most $H$ and obtain upper and lower bounds on the number of such matrices of a given rank and then apply them to count such matrices with a given determinant, or a given characteristic polynomial.

Statistics of ranks, determinants and characteristic polynomials of rational matrices

Abstract

We consider the set of matrices with rational entries having numerator and denominator of size at most and obtain upper and lower bounds on the number of such matrices of a given rank and then apply them to count such matrices with a given determinant, or a given characteristic polynomial.
Paper Structure (31 sections, 15 theorems, 244 equations)

This paper contains 31 sections, 15 theorems, 244 equations.

Table of Contents

  1. Introduction
  2. Set-up
  3. Motivation and approach
  4. Notation
  5. Main results
  6. Bounds for ${\mathbb Q}$
  7. Bounds for ${\mathbb Z}^{-1}$
  8. Preliminary results
  9. Bounds on some arithmetic functions
  10. Squares in arithmetic progressions
  11. Counting solutions to some equations and congruences
  12. Proof of Theorem \ref{['thm:rank Q']}
  13. Proof of Theorem \ref{['thm:cdn']}
  14. The case $n=2$
  15. The case $n\ge 3$
  16. ...and 16 more sections

Key Result

Theorem 2.1

For $n \ge m > r$ we have

Theorems & Definitions (22)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 12 more