Counting Elements on Full Matrix Groups over Finite Field with Prescribed Eigenvalues

TL;DR

This paper derives a formula to count matrices over finite fields with specified eigenvalues and applies it to establish bounds on the number of certain potent elements in finite rings.

Contribution

It introduces a new formula for counting matrices with prescribed eigenvalues over finite fields and uses it to derive inequalities for potent elements in finite rings.

Findings

Derived a formula for counting matrices with prescribed eigenvalues over finite fields

Established an inequality for the number of (k+1)-potent elements in finite rings

Provided a method to analyze matrix properties over finite algebraic structures

Abstract

In the present article we shown a formula to compute the number of all matrices over the finite field $F$ whit prescribed eigenvalues. Using this formula we obtain one inequality for the number of $(k+1)$-potent elements over finite rings.