Factorization of polynomials with analytic coefficients

TL;DR

This paper investigates polynomials with analytic coefficients, demonstrating that such polynomials can be factorized into linear factors within the same class, especially those related to matrix characteristic polynomials.

Contribution

It extends Newton's result to polynomials with analytic coefficients, showing they can be fully factorized into linear factors within the same analytic class.

Findings

Polynomials with analytic coefficients can be factorized into linear factors within the same class.

Characteristic polynomials of matrices with analytic entries are included in this factorization.

The result generalizes Newton's theorem to a broader class of polynomials.

Abstract

We study monic univariate polynomials whose coefficients are analytic functions of a real variable and whose roots lie in a specified analytic curve. These include characteristic polynomials of unitary and hermitian matrices whose entries are analytic functions. We use a result of Newton to prove that every polynomial in such a class is a product of degree one polynomials in the class.