Estimating irreducible-characteristic-polynomial matrices in higher dimensions via character sums

Develop a character-sum approach to estimate the number of n×n matrices over the finite field F_p with irreducible characteristic polynomials and entries in the interval [1,N], for n≥3.

Background

The paper proves an asymptotic formula for the number of 2×2 matrices over F_p whose entries lie in [1,N] and whose characteristic polynomials are irreducible, using bounds for tridimensional character sums with polynomial arguments. In the comments, the authors propose extending analogous questions to n×n matrices with n≥3.

For higher dimensions, irreducibility of the characteristic polynomial is not directly represented by a quadratic character condition, so the character-sum method used for 2×2 matrices does not presently provide a feasible estimation strategy. The unresolved problem is therefore to devise such a character-sum approach for counting higher-dimensional matrices with irreducible characteristic polynomials and short-interval entries.