Pairs of matrices with simple spectrum

TL;DR

This paper proves the Curto-Herrero conjecture for pairs of matrices with one having a simple spectrum, showing they can be distinguished by ranks of non-commutative polynomials and providing bounds on polynomial degrees.

Contribution

It establishes the two-sided Curto-Herrero conjecture for such matrix pairs and offers bounds on the degrees of relevant non-commutative polynomials.

Findings

Pairs with a simple spectrum are separated by ranks of non-commutative polynomials

The paper provides upper bounds on the degrees of these polynomials

The conjecture is proven in the two-sided case for this class of matrices

Abstract

We established two-sided Curto-Herrero conjecture for pairs of matrices, where the first matrix has a simple spectrum. Namely, it is shown that these pairs are separated by ranks of non-commutative polynomials in matrices. Moreover, we provided some upper bound on degrees of non-commutative polynomials which should be considered.