Polynomials with symmetric zeros

Abstract

Polynomials whose zeros are symmetric either to the real line or to the unit circle are very important in mathematics and physics. We can classify them into three main classes: the self-conjugate polynomials, whose zeros are symmetric to the real line; the self-inversive polynomials, whose zeros are symmetric to the unit circle; and the self-reciprocal polynomials, whose zeros are symmetric by an inversion with respect to the unit circle followed by a reflection in the real line. Real self-reciprocal polynomials are simultaneously self-conjugate and self-inversive so that their zeros are symmetric to both the real line and the unit circle. In this survey, we present a short review of these polynomials, focusing on the distribution of their zeros.

Keywords: Self-inversive polynomials, self-reciprocal polynomials, Pisot and Salem polynomials, Möbius transformations, knot theory, Bethe equations

References in corpus (9)

• Survey article: Seventy years of Salem numbers
• Solving and classifying the solutions of the Yang-Baxter equation through a differential approach. Two-state systems
• On the number of roots of self-inversive polynomials on the complex unit circle
• A seventeenth-order polylogarithm ladder
• Where are the roots of the Bethe Ansatz equations?
• On zeros of self-reciprocal polynomials
• How to count the number of zeros that a polynomial has on the unit circle?
• The necessary and sufficient condition for an algebraic integer to be a Salem number
• Polynomial Roots and Calabi-Yau Geometries

Cited by in corpus (4)

• How to count the number of zeros that a polynomial has on the unit circle?
• Dragging the roots of a polynomial to the unit circle
• Orthogonal polynomials and Möbius transformations
• The reciprocal algebraic integers having small houses