Implementation of Pellet's theorem

TL;DR

This paper enhances Pellet's theorem by providing explicit conditions for the existence of roots of the auxiliary polynomial and efficient methods for their computation, applicable to both scalar and matrix polynomials.

Contribution

It introduces explicit criteria and computational techniques for roots of the auxiliary polynomial in Pellet's theorem, improving its practical applicability.

Findings

Derived explicit conditions for auxiliary polynomial roots.

Proposed efficient algorithms for root computation.

Extended approach to matrix polynomial generalization.

Abstract

Pellet's theorem determines when the zeros of a polynomial can be separated into two regions, based on the presence or absence of positive roots of an auxiliary polynomial, but does not provide a method to verify its conditions or to compute the roots of the auxiliary polynomial when they exist. We derive an explicit condition for these roots to exist and, when they do, propose efficient ways to compute them. A similar auxiliary polynomial appears for the generalized Pellet theorem for matrix polynomials and it can be treated in the same way.