# Nearly Commuting Matrices

## TL;DR

This paper investigates the algebraic properties of pairs of matrices with diagonal commutators, proving $F$-purity and irreducibility under certain conditions, and discusses conjectures on their singularities across different characteristics.

## Contribution

It establishes $F$-purity and irreducibility results for algebraic sets of matrix pairs with diagonal commutators, extending understanding across various matrix sizes and characteristics.

## Key findings

- Proves $F$-purity of the algebraic set for 3x3 matrices.
- Shows the algebraic set is reduced and irreducible in any characteristic.
- Analyzes singularities and parameters of the coordinate rings.

## Abstract

We prove that the algebraic set of pairs of matrices with a diagonal commutator over a field of positive prime characteristic, its irreducible components, and their intersection are $F$-pure when the size of matrices is equal to 3. Furthermore, we show that this algebraic set is reduced and the intersection of its irreducible components is irreducible in any characteristic for pairs of matrices of any size. In addition, we discuss various conjectures on the singularities of these algebraic sets and the system of parameters on the corresponding coordinate rings.