Null ideals of sets of $3 \times 3$ similar matrices with irreducible characteristic polynomial

TL;DR

This paper investigates the null ideals of sets of 3x3 matrices over a field with an irreducible characteristic polynomial, establishing conditions for when these sets are core, and explores related algebraic and combinatorial structures.

Contribution

It provides new sufficient conditions for sets of 3x3 matrices to be core, especially over finite fields, and connects these results to block Vandermonde matrices and graph theory.

Findings

If the field is finite with q elements and |S| ≥ q^3 - q^2 + 1, then S is core.

The study links null ideals to properties of block Vandermonde matrices.

Results include insights into invertible matrix commutators and graph structures via difference relations.

Abstract

Let $F$ be a field and $M_n(F)$ the ring of $n \times n$ matrices over $F$. Given a subset $S$ of $M_n(F)$, the null ideal of $S$ is the set of all polynomials $f$ with coefficients from $M_n(F)$ such that $f(A) = 0$ for all $A \in S$. We say that $S$ is core if the null ideal of $S$ is a two-sided ideal of the polynomial ring $M_n(F)[x]$. We study sufficient conditions under which $S$ is core in the case where $S$ consists of $3 \times 3$ matrices, all of which share the same irreducible characteristic polynomial. In particular, we show that if $F$ is finite with $q$ elements and $|S| \geqslant q^3-q^2+1$, then $S$ is core. As a byproduct of our work, we obtain some results on block Vandermonde matrices, invertible matrix commutators, and graphs defined via an invertible difference relation.