# RANK OF THE DERIVATIVE OF THE PROJECTION TO SYMMETRIZED POLYDISC

## DOI:

https://doi.org/10.18173/2354-1059.2024-0016## Keywords:

Nevanlinna-Pick, interpolation, symmetrized polydisc## Abstract

The projection, also called the symmetrization mapping, from spectral ball to symmetrized polydisc is closely related to the spectral Nevanlinna-Pick interpolation problem. We prove that the rank of the derivative of the projection from the spectral unit ball to the symmetrized polydisc is equal to the degree of the minimal polynomial of the matrix at which we take the derivative. Therefore, it explains why the corresponding lifting problem is easier when the matrix base-point is cyclic since it is a regular point of the symmetrization mapping in the differential sense.

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