Implementation of Pellet's theorem
Provides practical computational tools for applying Pellet's theorem to polynomial root separation, benefiting numerical analysis and polynomial computation.
The authors derive explicit conditions for the existence of positive roots in the auxiliary polynomial of Pellet's theorem and propose efficient computation methods, extending the approach to matrix polynomials.
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.