Geometric aspects of Pellet's and related theorems

Abstract

Pellet's theorem determines when the zeros of a polynomial can be separated into two regions, according to their moduli. We refine one of those regions and replace it with the closed interior of a lemniscate that provides more precise information on the location of the zeros. Moreover, Pellet's theorem is considered the generalization of a zero inclusion region due to Cauchy. Using linear algebra tools, we derive a different generalization that leads to a sequence of smaller inclusion regions, which are also the closed interiors of lemniscates.