Local-global principle for triangularizability and diagonalizability of matrices

TL;DR

This paper establishes conditions under which matrices over number fields satisfy the local-global principle for triangularizability and diagonalizability, and identifies the stratified Brauer--Manin obstruction as the only obstacle in certain cases.

Contribution

It proves the local-global principle for these properties over principal ideal domains and characterizes the sole obstruction via the stratified Brauer--Manin obstruction in specific scenarios.

Findings

Local-global principle holds when $\\mathcal{O}_k$ is a principal ideal domain.

Stratified Brauer--Manin obstruction is the only obstacle in some cases.

Conditions for triangularizability and diagonalizability over number fields.

Abstract

Given a number field $k$ with the ring of integers $\mathcal{O}_k$ and a matrix $M\in \mathrm{M}_{n}(\mathcal{O}_k)$. We prove that if $\mathcal{O}_k$ is a principal ideal domain, the local-global principle for triangularizability and diagonalizability of $M$ holds. To explain the possible failures of the local-global principle, we prove that the stratified Brauer--Manin obstruction is the only obstruction to the local-global principle for triangularizability and diagonalizability of $M$ in some special cases.