Polynomials, roots, and interlacing

Abstract

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.

784 pages. Third part revised, new chapter on polynomial sequences satisfing a recurrence relation with polynomial coefficients. New sections: the division algorithm for interlacing polynomials, the Faa di Bruno problem, musings on the BMV conjecture, generators for 2 by 2 matrices of determinant 1 with interlacing entries, variations on multiaffine polynomials

Cited by in corpus (25)

• Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees
• Interlacing Families II: Mixed Characteristic Polynomials and the Kadison-Singer Problem
• Ramanujan Coverings of Graphs
• Interlacing Ehrhart Polynomials of Reflexive Polytopes
• $q$-log-convexity from linear transformations and polynomials with only real zeros
• The separating semigroup of a real curve
• Generalized Hurwitz polynomials
• Elements of Polya-Schur theory in finite difference setting
• Generalized Hurwitz matrices, generalized Euclidean algorithm, and forbidden sectors of the complex plane
• A black box method for solving the complex exponentials approximation problem
• Stable regions of Turán expressions
• Noncoherent Short-Packet Communication via Modulation on Conjugated Zeros
• Self-interlacing polynomials
• Maximal univalent disks of real rational functions and Hermite-Biehler polynomials
• Towards Constructing Ramanujan Graphs Using Shift Lifts
• Linear finite difference operators preserving Laguerre-Pólya class
• Dragging the roots of a polynomial to the unit circle
• Monomial to ultraspherical basis transformation and the zeros of polynomials
• Hermite-Poulain theorems for linear finite difference operators
• Problems around polynomials - the good, the bad and the ugly ...
• Spectrahedral relaxations of hyperbolicity cones
• Stability of combinatorial polynomials and its applications
• A Representation Theoretic Explanation of the Borcea-Brändén Characterization
• Strictly Real Fundamental Theorem of Algebra
• Diagonal Differential Operators