Seminar on Numerical Analysis

Recent extensions of scalar and matrix polynomial properties and their interactions

by Aaron Melman (Santa Clara University, USA)

Aula Riunioni (Dipartimento di Matematica)

Aula Riunioni

Dipartimento di Matematica


Several results for scalar polynomials are extended and then generalized to matrix polynomials. Among them are a directional version of Pellet's theorem for both scalar
and matrix polynomials, which establishes an exclusion interval for the magnitudes of polynomial zeros or eigenvalues having a given argument, as well as results involving real polynomials with sign restrictions on their coefficients. The latter lead to exclusion sectors
for polynomial eigenvalues.


In addition, it is shown how new results can be obtained by embedding scalar polynomials into a matrix polynomial framework, and examples are given of how the interaction
between scalar and matrix polynomials benefits both. If time (and interest) permits, an improvement of theorems by Hayashi and Hurwitz from the beginning of the twentieth
century will be presented. These theorems concern the location of polynomial zeros when the coefficients exhibit certain sign patterns.