Polynomials, roots, and interlacing

Details Polynomials, roots, and interlacing Polynomials, roots, and interlacing

2006-12-28

arxiv.org/abs/math/0612833v2

Mathematics

Classical Analysis and ODEs

784 pages. Third part revised, new chapter on polynomial sequences satisfing a recurrence relation with polynomial coefficient

Scientific paper

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.

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