Modified Gershgorin Disks for Companion Matrices

All the zeros of a polynomial are contained in the union of Gershgorin disks derived from its companion matrix, a consequence of Gershgorin's theorem. However, this theorem does not exploit the structure of the companion matrix. We will use this structure to obtain smaller zero inclusion regions, thereby providing some nonstandard results to accompany and illustrate this frequently covered topic in numerical and matrix analysis.