  This work is divided into three parts. The first part concerns polynomials in
one variable with all real roots. We consider linear transformations that
preserve real rootedness, as well as matrices that preserve interlacing. The
second part covers polynomials in several variables that generalize polynomials
with all real roots. We introduce generating functions and use them to
establish properties of a linear transformation. We also consider matrices and
matrix polynomials. The third part considers polynomials with complex roots.
The two main classes considered are polynomials with all roots in the left half
plane (stable polynomials) and those with all roots in the lower half plane
(Upper half plane polynomials). These naturally generalize to polynomials in
many variables.
  And, of course, there is much more.
