Properties of determinantal polynomials of subspaces of matrices over a finite field

TL;DR

This paper explores the properties of determinantal polynomials associated with subspaces of matrices over finite fields, linking algebraic geometry tools like Lang-Weil and Chevalley's theorems to matrix determinant behavior.

Contribution

It introduces a detailed analysis of the determinantal polynomial P_M for subspaces of matrices over finite fields, especially when the dimension is prime, and examines cases where non-zero determinant elements form a proper subspace.

Findings

The determinantal polynomial P_M is homogeneous of degree n in d variables.

Theorems relating the zeros of P_M to the properties of matrix subspaces over finite fields.

Examples showing non-zero determinant elements can form a proper subspace.

Abstract

Let K be a field and let M_n(K) denote the space of n x n matrices with entries in K. Let M be a subspace of M_n(K) of dimension d with the property that there are elements in M with non-zero determinant. Given a basis of M, we define the determinantal polynomial P_{M} of M with respect to the basis. It is a homogeneous polynomial of degree n in d indeterminates that gives the determinant of any element of M by evaluation in K^d. This paper investigates the interrelationship of M and P_{M}. We confine ourselves to finite fields K, where we can obtain useful information by applying the Lang-Weil theorem on the number of zeros of absolutely irreducible polynomials. A combination of Chevalley's theorem on the zeros of polynomials in several variables and the Lang-Weil theorem leads to theorems about the characteristic polynomials of elements of M when n is a prime. We also draw attention to cases when the elements of M with non-zero determinant are a proper subspace, and provide non-trivial examples of this phenomenon.