Rank of the derivative of the projection to symmetrized polydisc

TL;DR

This paper establishes a precise relationship between the rank of the derivative of a projection from the spectral unit ball to the symmetrized polydisc and the degree of the minimal polynomial of the matrix involved.

Contribution

It provides a novel result linking the derivative's rank to the minimal polynomial degree in the context of spectral and symmetrized polydisc projections.

Findings

Rank of the derivative equals the minimal polynomial degree.

The result applies to matrices at points where the derivative is considered.

It advances understanding of the geometric structure of spectral and symmetrized polydisc mappings.

Abstract

We prove that the rank of the derivative of the projection from spectral unit ball to symmetrized polydisc is equal to the degree of the minimal polynomial of the matrix at which we take derivative.